Ohanian Fisica 3e Capitulo Muestra 22

PArTE

4

Electricidad y magnetismo

CONTENIDOCAPTULO 22 CAPTULO 23 CAPTULO 24 CAPTULO 25 CAPTULO 26 CAPTULO 27 CAPTULO 28 CAPTULO 29 CAPTULO 30 CAPTULO 31 CAPTULO 32 Fuerza elctrica y carga elctrica El campo elctrico La ley de Gauss Potencial electrosttico y energa Capacitores y dielctricos Corrientes elctricas y la ley de Ohm Circuitos de corriente directa Fuerza y campo magntico Cargas y corrientes en campos magnticos Induccin electromagntica Circuitos de corriente alterna

Un microchip contiene circuitos diseados con precisin, empacados en un espacio muy pequeo. Este microalambre de oro es una de varias conexiones al microchip, y lleva a terminales ms grandes que se usan para insertar el microchip en una tarjeta de circuito, como las que se encuentran en el interior de una computadora.

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CAPTULO

22

Fuerza elctrica y carga elctrica

CONCEPTOS EN CONTEXTO22.1 La fuerza electrosttica 22.2 Ley de Coulomb 22.3 La superposicin de fuerzas elctricas 22.4 Cuantizacin y conservacin de la carga 22.5 Conductores y aislantes; carga por friccin o por induccin

Las fotocopiadoras y las impresoras lser usan modernas partculas de tner en tonos magenta, cian, amarillo y negro, hechas en forma esfrica, con un recubrimiento de polmero. Una fuerza elctrica de atraccin las mantiene sujetas a la placa detrs de ellas; tambin las partculas ejercen fuerzas elctricas de repulsin entre s. Con las propiedades de la fuerza elctrica y la carga elctrica que se presentan en este captulo, ser posible contestar preguntas como:

Conceptos en contexto

? Cul es la fuerza que ejerce una partcula de tner sobre otra? ? Cul es la fuerza total sobre una partcula rodeada por varias otras ? Cmo se transfieren las partculas de tner para formar una imagen?(La fsica en la prctica. Xerografa, pgina 709) partculas? (Ejemplo 7, pgina 704) (Ejemplo 4, pgina 701)

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22.1

La fuerza electrosttica

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a sociedad humana depende de la electricidad. Con una falla de la energa elctrica se demuestra esa dependencia: se paran los trenes subterrneos, las luces de los semforos, el alumbrado de las calles y las luces de los hogares; los refrigeradores dejan de funcionar; no se puede cocinar; la casa no puede calentarse; no funcionan radios, televisores ni computadoras. Pero la dependencia de la electricidad es ms profunda que lo que sugiere la dependencia de maquinaria y utensilios elctricos. La electricidad es un ingrediente esencial de todos los tomos en el organismo humano y en el medio ambiente. Las fuerzas que mantienen unidas las partes de un tomo son fuerzas elctricas. Adems, tambin son elctricas las fuerzas que unen los tomos para conformar una molcula, y unen a gran escala las molculas, elementos constructivos que forman estructuras macroscpicas como rocas, rboles, el cuerpo humano, rascacielos y los grandes buques-tanque. Todas las fuerzas mecnicas de contacto de la cotidianidad, como el empuje de una mano contra una puerta, la tensin de un cable de elevador, la presin del agua contra el casco de un barco, no son ms que fuerzas elctricas combinadas sobre muchos tomos. Entonces, el entorno inmediato est dominado por fuerzas elctricas. En los captulos que siguen se estudiarn las fuerzas elctricas y sus efectos. Para comenzar (captulos 22 a 28) se supondr que las partculas que ejercen esas fuerzas estn en reposo, o que slo se mueven muy despacio. Las fuerzas elctricas que se ejercen bajo estas condiciones se llaman fuerzas electrostticas. Despus (captulos 29 a 31) se examinarn las fuerzas elctricas cuando las partculas se mueven con velocidad uniforme o casi uniforme. Bajo estas condiciones, se modifican las fuerzas elctricas; adems de la fuerza electrosttica se produce una fuerza magntica, que depende de las velocidades de las partculas. Las fuerzas electrostticas y magnticas combinadas se llaman fuerzas electromagnticas. Por ltimo, se examinarn las fuerzas que se ejercen cuando las partculas se mueven con movimiento acelerado (captulo 33). En ese caso, las fuerzas electromagnticas se vuelven a modificar con una consecuencia drstica, que es la emisin de ondas electromagnticas, como son las ondas luminosas y las radioelctricas. La electricidad se descubri a travs de la friccin. Los antiguos griegos notaron que cuando se frotaban barras de mbar (elektron, en griego) contra un pedazo de tela o de piel, emitan chispas y atraan pequeos trozos de paja o plumas. Se puede repetir fcilmente este antiguo descubrimiento si se frota un peine de plstico en una camisa o un suter: en la oscuridad se puede ver que se produce una multitud de pequeas chispas con este frotamiento, y el peine electrificado atrae pequeos trozos de papel o de tela. En el siglo xix se desarrollaron en forma gradual las aplicaciones prcticas de la electricidad, pero slo fue hasta el siglo xx que se reconoci la dominante presencia de las fuerzas elctricas que mantienen unida toda la materia en el entorno.

L

fuerza elctrica

fuerza magntica

22.1 LA FUErZA ELECTrOSTTICALa materia ordinaria slidos, lquidos y gases est compuesta de tomos, cada uno de ellos con un ncleo rodeado por multitud de electrones. Por ejemplo, la figura 22.1 muestra la estructura de un tomo de nen. En el centro de este tomo hay un ncleo formado por diez protones y diez neutrones, empacados en forma muy apretada; el dimetro del ncleo slo mide unos 6 1015 m. En torno a este ncleo se mueven diez electrones, que estn confinados en una regin aproximadamente esfrica de unos 3 1010 m de dimetro. El tomo se parece un poco al sistema solar, donde el ncleo est representado por el Sol, y los electrones por los planetas. En el sistema solar, la fuerza que mantiene a un planeta cerca del Sol es gravitacional. En el tomo, la fuerza que mantiene a un electrn cerca del ncleo es la fuerza elctrica de atraccin entre l y los protones en el ncleo. Esta fuerza elctrica se parece a la gravitacional, porque disminuye en proporcin al

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CAPTULO 22


El nen tiene diez electrones

y su ncleo tiene diez protones.

FIGUrA 22.1 tomo de nen. Se ven los electrones en un instante en el tiempo, como se veran en un microscopio (hipottico) extremadamente poderoso.

Los electrones estn principalmente dentro de una esfera pequea centrada en el ncleo.

carga elctrica

coulomb (C)

cuadrado inverso de la distancia. Pero la fuerza elctrica es mucho mayor que la fuerza gravitacional. La atraccin elctrica entre un electrn y un protn (a cualquier distancia) es aproximadamente 2 1039 veces mayor que la atraccin gravitacional. As, la fuerza elctrica es, con mucho, la mxima fuerza que acta sobre un electrn en un tomo. La otra gran diferencia entre la fuerza gravitacional y la fuerza elctrica es que la primera, entre dos partculas, siempre es de atraccin, mientras que las fuerzas elctricas pueden ser de atraccin, de repulsin o nulas, dependiendo 1010 m de qu partculas se consideren. La tabla 22.1 presenta un resumen cualitativo de las fuerzas elctricas entre las partculas fundamentales. Se dice que las partculas que ejercen fuerzas elctricas tienen una carga elctrica, y que las que no ejercen fuerzas elctricas no la tienen. As, se puede percibir la carga elctrica como la fuente de la fuerza elctrica, exactamente como la masa es la fuente de la fuerza gravitacional. Los electrones y los protones tienen carga elctrica, pero los neutrones no. La fuerza entre un electrn y un protn, la fuerza entre un electrn y un electrn, y la fuerza entre un protn y un protn tiene siempre la misma magnitud (para una distancia determinada). As, las intensidades de las fuerzas elctricas asociadas con los electrones y los protones tienen magnitudes iguales; esto es, sus cargas elctricas tienen magnitudes iguales. Para la formulacin matemtica de la ley de la fuerza elctrica, se asigna al protn una carga positiva, y al electrn una carga negativa (las dos de igual magnitud). A esas cargas del protn y del electrn se les representa por +e y e, respectivamente. En la tabla 22.2 se resumen los valores de esas cargas. En funcin de estas cargas elctricas, entonces ser posible declarar que la fuerza elctrica entre cargas de signos similares es de repulsin, y la fuerza elctrica entre cargas de signo distinto es de atraccin. El valor numrico de la carga e del protn depende del sistema de unidades. En el sistema de unidades SI, la carga se expresa en coulombs (C), y los valores numricos correspondientes de las cargas fundamentales del protn y del electrn son* e = 1.60 1019 C para el protn

carga del protn y el electrn

e = 1.60 1019 C para el electrn

(22.1)

TABLA 22.1 FUErZAS ELCTrICAS (CUALITATIVAS)PARTCULAS Electrn y protn Electrn y electrn Protn y protn Neutrn y cualquier cosa FUERZA Atraccin Repulsin Repulsin Cero

TABLA 22.2 CArGAS ELCTrICAS DE PrOTONES, ELECTrONES Y NEUTrONESPARTCULA Protn, p Electrn, e Neutrn, n CARGA +e e 0

* Como en todas las constantes fsicas, se han redondeado a tres decimales esos valores. En el apndice 6, o en el interior de las cubiertas del libro, se encuentra un valor ms exacto.

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22.1

22.1

La fuerza electrosttica

The Electrostatic Force

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697

In the SI system of units, the coulomb is defined in terms of a standard electric En el sistema de unidades SI se define el coulomb en funcin amount of eleccurrent, that is, a standard rate of flow of charge: one coulomb is the de una corriente elctrica estndar,current of one ampere delivers in one second. Unfortunately, thecantitric charge that a esto es, una tasa de flujo estndar de cargas: un coulomb es la defdad de carga elctrica que unainvolves the use ofampere produce andun segundo. Desinition of the standard current corriente de un magnetic fields, en we will therefore afortunadamente, thedefinicin of the precise definition of implica el uso de campos have to postpone la question de la corriente estndar ampere and coulomb to a magnticos, por lo que se tendr que posponer para un captulo posterior la definicin later chapter. precisa del ampere ofdel coulomb. One coulomb y charge represents a large number of fundamental charges; this is Unseen by taking the inverse of Eq. (22.1), cantidad de cargas fundamentales. Eso easily coulomb de carga representa una gran se aprecia con facilidad sacando la inversa de la ecuacin (22.1), 1C 1 1.60 1019 e 6.25 1018 e

As, un coulomb equivale a ms de 6 1018 cargas fundamentales. Como la carga funThus, a coulomb is more than 6 billion billion fundamental charges. Since the fundadamental es tan pequea, con frecuencia se puede pasar por alto su carcter discreto, en mental charge so small, we can often ignore the discrete character of considerar que la aplicaciones deiselectricidad en la prctica y la ingeniera, y se puedecharge in practical and engineering applications of electricity, continua. treat macroscopic charge distridistribucin de las cargas macroscpicas esand we canEsto es anlogo a considerar que butions as continuous. This is analogous son continuas, si bien mass distributions as las distribuciones macroscpicas de masato treating macroscopiclo cierto es que en una continuous, even though, on a microscopic scale, the discretos. escala microscpica la masa est formada por tomosmass consists of discrete atoms. The net elctrica neta of a cuerpo que contenga number of electrons and proLa carga electric chargede unbody containing some cierta cantidad de electrones y tons is the (algebraic) (algebraica) electron and proton charges. For instance, elctrica de protones es la suma sum of the de las cargas de ellos. Por ejemplo, la carga the net electric charge of an contenga cantidades iguales de of electrons and protons is zero; neta de un tomo queatom containing equal numberselectrones y protones es cero, esto that tomo es elctricamente neutro. En algunas ocasiones los tomos pierden un es, el is, the atom is electrically neutral. Sometimes atoms lose an electron, and sometimes they gain an extra un electrn adicional. Los tomos con electrones faltantes o con electrn, y en otras, gananelectron. Such atoms with missing electrons or with extra elecelectrones called ions. se llaman iones. positive charge if they have lost electrons, and a trons are adicionales They have a net Tienen una carga neta positiva si han perdido electrones, y una carga they have gainedhan ganado electrones. La carga positiva o nenet negative charge if neta negativa si electrons. The positive or negative charge on gativa en un cuerpo macroscpico, como por ejemplo un peinerubbingarises in the a macroscopic bodysuch as on a plastic comb electrified by de plstico electrizado por frotamiento,ase produce de an misma of electrons. una deficiencia o un exceso de same way, from deficiency or la excess manera, por electrones. The electric forces between two neutral atoms tend to cancel; each electron in one Las attracted by the entre dos the nucleus of tienden anularse; simultaneously atom isfuerzas elctricasprotons in tomos neutrosthe otheraatom, and cada electrn en un is repelledatrado equal number of electrons of that otro tomo, y al mismo tiempo es it tomo es by the por los protones en el ncleo del atom. However, the cancellation repelido electric attractive and repulsive forces among the electrons andSin embargo, a of these por una cantidad igual de electrones de este ltimo tomo. the protons in veces no es total la anulacin de estas fuerzas de atraccin y repulsin entre electrones the two atoms is sometimes not complete. For instance, the contact force between y protones de los dos tomos. Por ejemplo, la fuerza de contacto entre dos tomos two atoms close together arises from an incomplete cancellation of the attractive and cercanos se debe a una anulacin incompleta de las fuerzas de atraccin y repulsin. La repulsive forces. The force between the atoms depends on the relative locations of the fuerza entre los tomos depende de las ubicaciones relativas de los electrones y los electrons and the nuclei. If the distributions of the electrons are somewhat distorted ncleos. Si se distorsionan algo las distribuciones de los electrones, de modo que en so, on the average, the en un tomo estn ms closer to los electrones the tomo promedio los electroneselectrons in one atom arecercanos athe electrons ofen elneighboring atom su ncleo, la fuerza neta entre force between these repulsin. La figura vecino que a than to its nucleus, then the net esos tomos ser de atoms will be repulsive. muestra una distorsin distortion that leads to a neta de repulsin; la distorsin 22.2aFigure 22.2a shows such a que produce una fuerzarepulsive net force; the distortion may ser intrnseca de la the structure tomo, o inducida por la the presence tomo puedeeither be intrinsic to estructura delof the atom or induced by presencia del of the neighboring atom. Figure 22.2b shows a distortion that leads to an attractive force. vecino. La figura 22.2b muestra una distorsin que produce una fuerza de atraccin. Likewise, las fuerzas elctricas entre dos cuerpos macroscpicos neutrales separaAsimismo,the electric forces between two neutral macroscopic bodies separated by some appreciable distance tend to tienden a cancelarse. Por ejemplo, si bodies are dos por una distancia considerablecancel. For example, if the macroscopiclos cuerpos a baseball and tennis ball separated by a y una de tenis then each electron distancia de macroscpicosason una pelota de bisboldistance of 2 m,separadas por una of the basedos metros, cada by the protonspelota de bisbol esbut simultaneously it is repelled by ball is attracted electrn de la of the tennis ball, atrado por los protones de la pelota deelectrons of the tennis tiempo esthese forces cancel each other. Only when the sur-y the tenis, pero al mismo ball; and repelido por los electrones de la pelota de tenis, estas fuerzastwo balls are entrenear onecuando las superficies willlas dos pelotas estn lo faces of the se cancelan very s. Slo another (touching) de the atoms in one sursuficientemente electric force on thosetomosother surface. face exert a net cerca (se tocan), los in the en una superficie ejercern una fuerza elctrica neta en los de la otra superficie. between neutral macroscopic bodies explains This cancellation of the electric forces Esta cancelacin de electric attractions entre cuerpos macroscpicos neutrales why we do not see largelas fuerzas elctricasor repulsions between the macroscopic explica porour environment, even though the electric forces between individual elecbodies in qu no se observan grandes fuerzas de atraccin o repulsin entre ellos en el medio, aunque las fuerzas elctricas than the gravitational forces. Most such macrotrons and protons are much stronger entre electrones y protones sean ms fuertes que las fuerzas de are electrically neutral, and they therefore will exert noson elctricamente scopic bodies gravedad. Muchos de estos cuerpos macroscpicos net electric forces neutros, y por tanto no ejercen fuerza elctrica neta en los dems, salvo por fuerzas de on each other, except for contact forces. contacto.

ion ionLas cargas del mismo signo estn ms cerca en promedio. a)

ncleo b)

Las cargas de signos contrarios estn ms cerca en promedio.

FIGUrA 22.2 a) Dos tomos vecinos distorsionados. Las regiones de color reFIGURE 22.2 (a) Two neighboring dispresentan la distribucin promedio de los torted atoms. The colored regions represent electrones. Los electrones del tomo de the average distribution of the electrons. la izquierda estn ms cerca de la closer to The electrons of the left atom aremayor parte de los the electrons tomo de atom than most ofelectrones delof the rightla derecha que nucleus. This ste. Eso causa una to its del ncleo de results in a net repulsive fuerza neta de repulsin entre los tomos. force between the atoms. (b) The electrons b) Los electrones closer to the la izquierda of the left atom aredel tomo denucleus of estn ms cerca del ncleo del tomo de la the right atom than to most of its electrons. derecha que a la attractive force between This results inde netmayor parte de los electrones de the atoms. ste. Eso causa una fuerza neta de atraccin entre los tomos.

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698 698

CAPTULO 22

CHAPTER 22 Electric Force and Electric Charge CHAPTER 22 Electric Force and Electric Charge


CHARLES AUGUSTIN de COULOMB CHARLES AUGUSTIN de COULOMB (1736 1806) French physicist, shown here CHArLES AUGUSTIN de COULOMB (1736 1806) French physicist, shown here with the torsion balance, which he se ve con (1736-1806) Fsico francs; aquinvented. with the torsion balance, which he invented. He established that the electric force Establela balanza de torsin, que l invent.between He established that the electric force between small la fuerza elctrica entre esferas cargaci quecharged balls obeys an inverse-square small charged balls obeys an inverse-square law. das pequeas obedece a una ley del inverso del law. cuadrado.

tric forces on each other, but only insignificant gravitational forces. Explain this dad, forces on each other, butfuerzas elctricas. Los electrones forces. tomo ejercen tric pero slo insignificantes only insignificant gravitational en un Explain this difference. grandes fuerzas elctricas entre s, pero slo insignificantes fuerzas de gravedad. Explidifference. QUESTION 2: A stone of mass 1.0 kg rests on the ground. What is the net electric force que esa diferencia. QUESTION 2: A stone of mass 1.0 kg rests on the ground. What is the net electric force that the ground exerts on the stone? PrEGUNTA 2: Una piedra de 1.0 kg de masa descansa en el suelo. Cul es la fuerza that the ground exerts on the stone? elctrica neta que ejerce el piso sobre la piedra? QUESTION 3: Suppose that in Fig. 22.2a, the atomic nuclei were displaced toward each QUESTION 3: Suppose that in Fig. 22.2a, the atomic nuclei were displaced toward each other, instead Supngase que enother. Would thelos ncleos force between the atoms of away from each la figura 22.2a net electric atmicos se desplazaran PrEGUNTA 3: of away from each other. Would the net electric force between the atoms other, instead be attractiveotro, en lugar de alejarse entre s. La fuerza elctrica neta entre los tomos or repulsive? unoattractive or repulsive? be hacia el sera de atraccinelectrons are added to 1.0 coulomb of positive charge. The net charge QUESTION 4: Six o de repulsin? QUESTION 4: Six electrons are added to 1.0 coulomb of positive charge. The net charge is approximately agregan seis electrones a 1.0 coulomb de carga elctrica positiva. La PrEGUNTA 4: Se is approximately carga neta aproximada es5.0 C (A) 7.0 C (B) (C) 1.0 C (D) 6e (E) 5e (A) 7.0 C (B) 5.0 C (C) 1.0 C (D) 6e (E) 5e (A) 7.0 C (B) 5.0 C (C) 1.0 C (D) 6e (E) 5e

QUESTION 1: The planets in the Solar System exert large gravitational forces on each QUESTION 1: The planets in the Solar System exert large gravitational forces on each other, but only insignificant electric forces. The electrons in an atom exert large elecPrEGUNTA only insignificant electric forces. The electrons in grandes fuerzas de graveother, but 1: Los planetas del sistema solar ejercen entre s an atom exert large elec-

C h e c ku p 2 2 . 1 C h e c ku p 2 2 . 1 revisin 22.1

22.2 LEY DE COULOMB 22.2 COULOMB S LAW 22.2 COULOMB S LAWAs already mentioned above, the electric force between two particles decreases with Como ya sementioned above, the electric force between two particles decreases with As already dijo, la fuerza elctrica entre dos partculas disminuye con el inverso del the inverse square of the distance, just as does the gravitational force. The dependcuadrado de square of the distance, just as does the gravitational force. The dependthe inverse la distancia, tal como sucede en la fuerza gravitacional. El hecho de que la ence the electric force on distance was discovered through experiments by Charles fuerzaofelctrica dependa on distance was discovered through experimentsAugustin de ence of the electric force de la distancia fue descubierto por Charles by Charles Augustin de Coulomb, who investigated the repulsion between small balls that he Coulomb porCoulomb, who investigated the repulsion between small balls that he Augustin de medio de experimentos. Investig la repulsin entre pequeas esferas charged cargado por process. To measure the force between the balls, he used a dela rubbing frotacin. Para medir la que hababy a rubbing process. To measure the fuerzabetween the balls, he una delicada charged by force entre las esferas us used a delicate torsion balance (see Fig. 22.3) similar to the torsion balance later used by Henry balanza de torsin (vase Fig. 22.3) similar to the torsion balance later used by Henry la figura 22.3) parecida a la balanza que us despus Henry icate torsion balance (see Cavendish to measure gravitational forces. His experimental results are summarized Cavendish para medir las fuerzas deforces. His Sus resultadosresults are summarized Cavendish to measure gravitational gravedad. experimental experimentales se conin Coulombs Law: densan en la ley de Coulomb: in Coulombs Law: The magnitude of the electric force that a particle exerts on another particle is directly La magnitud deof the electric force that a ejerce una partcula sobre otra partcula es The magnitude la fuerza elctrica que particle exerts on another particle is directly proportional to the product of their charges and inversely proportional to the square directamente proporcional al producto de sus cargas, e inversamente proporcional al proportional to the product of their charges and inversely proportional to the square of the distance between que las separa. La of the force la fuerza line la lnea que cuadrado de la distancia them.The directiondireccin deis along thesiguejoining the of the distance between them.The direction of the force is along the line joining the particles. une a las partculas. particles. Mathematically, the electric force F that a particle of charge q exerts on a partiMatemticamente, laelectric elctricathat a particle una partculaexerts on aq sobre Mathematically, the fuerza force F F que ejerce of charge q de carga particle of charge deat a distance distancia r sethe formula la frmula una partcula q carga q a la r is given by calcula con cle of charge q at a distance r is given by the formula qq (22.2) (22.2) F k qq 2 F k r2 (22.2) r donde k es una constante de proporcionalidad. Esta frmula no slo da como resultado where k is a constant of proportionality. This formula not only gives the magnitude of where k is a de la fuerza, sino tambin la direccin, si not only gives the magnitude of la magnitud constant of proportionality. This formula se interpreta que un valor positivo the force, but also the direction, if we interpret a positive value of the force F as repulsive the fuerza F also repulsin, y un we interpret a positive value of the force F as repulsive de laforce, but como the direction, ifvalor negativo como atraccin. Por ejemplo, en el caso and a negative value as attractive. For instance, in the case of the force exerted by a proton de laafuerza que ejerce attractive. For instance, in the case of the son q= e y q = e, y con and negative value as un protn sobre un electrn, las cargas force exerted by a proton on an electron, the charges are q e and q e, and the formula (22.2) yields la frmula (22.2) secharges are q e and q e, and the formula (22.2) yields on an electron, the obtiene2 e (e ) e2 F k e (e ) k e 2 Fk k r 2 r2 r2 r

Online Online Concept Concept Tutorial Tutorial

24 24

b

a

FIGURE 22.3 Coulombs torsion balance. FIGUrA 22.3 Coulombs torsion balance. Balanza de torsin de FIGURE The beam carries a tiene charged ball Coulomb. La barra small una pequea(a) at The beam carries a small charged ball (a) at one end, and counterweight on the conesfera cargadaaa) en un extremo, y unother. one end, and a counterweight on the other. A second el otro. Una ball (b) esfera trapeso en small chargedsegunda is brought A second small charged ball (b) is brought near the first ball. If the balls carry charges cargada b) se acerca a la primera. Si las near the first ball. If the balls carry charges of equal signs, they repel each iguales, se esferas tienen cargas de signosother, and the of equal signs, they repel each other, and the repelen entrebalance rotates. de la beam of the s, y gira la barra beam of the balance rotates. balanza.

(22.3) (22.3) (22.3)

que es negativa, por lo que indica atraccin. which is negative, indicating attraction. which is negative, indicating attraction. La fuerza elctrica that the particle of charge carga q sobre la particle ofde cargaq The electric force que ejerce la partcula de q exerts on the partcula charge q The electric force that the particle of charge q exerts on su charge q tienethe misma magnitud as the force exertedq sobre q, perothe particle ofes opuesta. has la same magnitude que la ejercida por by q on q, but thedireccin direction. opposite has the samemutuas son un the force exerted by q (vasebut the opposite direction. magnitude as par de accin-reaccin on q, los dos ejemplos en la figuEstas fuerzas forces are an actionreaction pair (see Fig. 22.4 for two examples). These mutual These mutual forces are an actionreaction pair (see Fig. 22.4 for two examples). ra 22.4).

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F q'22.2 Coulombs Law Para cargas diferentes, la fuerza de Law 22.2 Coulombs Law 22.2 Coulombs 22.2 Coulombs Law 22.2 est a Coulomb atraccin sobre cada una Ley delo largo de la lnea que une a las cargasa) b)

699 699 699 699

699

q F

F q ry tambin la fuerza de repulsin, cuando las cargas son semejantes.

r

F q'Para cargas diferentes, la fuerza de atraccin sobre cada una est a lo largo de la lnea que une a las cargasb)

q' F

FIGURE 22.4 (a) Two charged particles q and q, one of which is positive and FIGURE 22.4 (a) Two charged particles q and q, one of which una negativa. FIGURE 22.4 (a) Two charged cargadas, and q, one of which is positive and FIGURE 22.4 (a) Two charged particles q and q, one of which is positive and FIGUrA 22.4 a) Dos partculasparticles q q y q, una positiva y is positive and one of which is negative. (b) Two other particles q and q; both are negative. one of which is negative. (b) Two other particles q and q; both are negative. one of which is negative. (b) Two other particles q and q; both are negative. b)one of which is negative.y(b) Two other particles q and q; both are negative. Otras dos partculas q q; ambas negativas. Fq

y tambin la fuerza En el sistema de unidades SI, a la constante de proporcionalidad kCoulombs Law, de la ley de CouIn the SI systemdeof units, the constant of proportionality k in Coulombs Law, r In the SI system of units, the constant of proportionality k in SI system of units, the constant of proportionality k repulsin, cuando las In thellama constante de Coulomb, o constante de la fuerza in Coulombs Law, lomb se le Coulombcargas son semejantes. elctrica; su valor es called the Coulomb constant, or the electric force constant, has the value called the constant, or the electric force constant, has the value called the Coulomb constant, or the electric force constant, has the value2 2 k 8.99 1099 N m22/C22 k 8.99 10 9 Nm2 /C2 k 8.99 109 Nm /C

(22.4) (22.4) (22.4) (22.4)

Coulomb constant constante de Coulomb Coulomb constant Coulomb constant Coulomb constant

This constant is traditionally written in the more complicated but equivalent form This constant is traditionally written in the forma ms complicada, pero equivalente: This constant is traditionally written in the more complicated but equivalent form Por tradicin, esta constante se expresa en la more complicated but equivalent formF

q'

1 1 k 1 k k 4p0 4p0 4p00

(22.5) (22.5) (22.5) (22.5)

with with siendo with12 0 8.85 1012 C22/(Nm22) 0 8.85 1012 C2/(Nm2) /(Nm ) 00 8.85 1012 C2/(Nm2)

(22.6) (22.6) (22.6) (22.6)

permittivity constant permittivity constant permittivity constant constante constant permittivityde permitividad

The quantity (epsilon nought) is called the electric constant or the permittivity The quantity (psilon cero) se llama constante elctrica, o constante de permitiThe quantity 0 (epsilon nought) is called the electric constant or the permittivity 0 La cantidad 000 (epsilon nought) is called the electric constant or the permittivity constant. In terms of the permittivity constant, Coulombs Law for the force that a constant. In terms of the permittivity constant, Coulombs Law for de force that a constant. In terms of the permittivity constant, Coulombs Law for the force that a vidad. En trminos de la constante de permitividad, la ley de Coulombthe la fuerza que particle of charge q exerts on a particle of charge q becomes particle of charge q exerts q sobre una partcula de carga q ejerce una partcula de cargaon a particle of charge q becomes se transforma en particle of charge q exerts on a particle of charge q becomes 1 qq 1 qq F 1 qq F F 4p0 r 22 2 4p0 r 2 4p00 r (22.7) (22.7) (22.7) (22.7)Coulombs Law ley de Coulomb Coulombs Law Coulombs Law Coulombs Law

Recurdese que unpositive value of F F representa una fuerza dedirected along the line valor positivo de repulsin dirigida a lo Remember that a positive value of F indicates a repulsive force directed along the line Remember that a a repulsive force Remember that a positive cargas, indicates a repulsive force directed along the de largo de thelnea que and avalue of Fyindicatesvalor negativo representaAlthough line la charges, une las que indicates an attractive force. una fuerza un joining the charges, and a negative value indicates an attractive force. Although the joining and a negative value indicates value the joining the charges, ecuacin (22.2), con el valor dean constante definido en la ecuaatraccin. Aunque with the negative the constant given attractive force. Although the la la expression (22.2), with the value of the constant given in Eq. (22.4), is most convenexpression (22.2), with the value of the constant given in Eq. (22.4), is most conven(22.2), value of in Eq. (22.4), is most convenexpression es ms cmoda para clculos numricos cin for the numerical calculation of the Coulomb force,de la fuerza de Coulomb, la (22.4), ient for the numerical calculation of the Coulomb force, the somewhat more compliient the somewhat more ient for the numerical calculation of the Coulomb force, the somewhat more compliecuacin (22.7), algo ms complicada, es la que se usa en fsica e ingeniera. compliEs claro cated expression (22.7) is widely used in physics and engineering. Of course, the two cated expression (22.7) is widely used in physicsequivalentes y producen los the two expression (22.7) is widely used in physics and engineering. Of course, the two and engineering. Of course, mismos cated que las dos ecuaciones son matemticamente expressions are mathematically equivalent, and they give the same results. expressions expressions are mathematically equivalent, and they give the same results. resultados. are mathematically equivalent, and they give the same results. Coulombs Law applies to particleselectrons and protonsand also to any small Coulombs Law applies to particleselectrons electrones y protones, y any small Coulombs Law applies to particleselectrons and protonsand also to tambin La ley de Coulomb se aplica a partculas, comoand protonsand also to any smalla charged bodies, provided that the sizes of these bodies are much smaller than the discharged cuerpos cargados pequeos, of these bodies tamaos de esos cuerpos sean charged bodies, provided that the sizes of these bodies are much smaller than the distodos losbodies, provided that the sizessiempre que los are much smaller than the distances between them; such bodies are called point charges. Equation (22.2) obviously tances menores que las distancias entre ellos; a esos cuerpos se les llama cargas punmucho between them; such bodies are called point charges. Equation (22.2) obviously tances between them; such bodies are called point charges. Equation (22.2) obviously resembles obvio que la ecuacin (22.2) se parece a la ley de Newton de la fuerza graviresembles Newtons Law for the gravitational force (see Section 9.1); the constant k tuales. Es Newtons Law for the gravitational force (see Section 9.1); the constant k resembles Newtons Law for the gravitational force (see Section 9.1); the constant k is analogous tola seccin 9.1); la constant G, and the electricconstante gravitacionalto is analogous to the gravitational constant k and the electric charges are analogous G, tacional (vase the gravitational constant G, and the electric charges are analogous to is analogous to the gravitational constanteG, es anloga a la charges are analogous to the gravitating masses. el equivalente de las masas que gravitan. the gravitating masses. ythe gravitating masses. las cargas elctricas son

point charge point charge point charge point charge carga puntual

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CHAPTER 22 Fuerza elctrica y carga elctrica CAPTULO22 Electric Force and Electric Charge CHAPTER 22 Electric Force and Electric Charge

E XAMPLE 1 EJEMPLO 1 1 EXAMPLE E XAMPLE

Compare the magnitudes of the gravitational force of attracComprense las magnitudes de la gravitational force of attracCompare the magnitudes of the fuerza gravitacional de atraction y deof the electric force of attraction between the electron and la fuerza elctrica de atraccin entre el electrn y el cin and of the electric force of attraction between the electron tion and the proton in a hydrogen atom. According to Newtonian mechanics, what is protn de un tomo de hidrgeno. De acuerdo con Newtonian mechanics, what is and the proton in a hydrogen atom. According to la mecnica de Newton cul es the acceleration of the electron? Assume that the distance between these particles la aceleracin delof the electron? Assume que la distancia entre estas partculas, en the acceleration electrn? Si se supone that the distance between these particles in a hydrogen atom is 5.3 1011 m. un tomo de hidrgeno, es 1011 11 m. in a hydrogen atom is 5.3 5.3 10 m. SOLUTION: From Chapter el captulo 9, la magnitud de la fuerza gravitacional es SOLUCIN: De acuerdo con9, the magnitude of the gravitational force is SOLUTION: From Chapter 9, the magnitude of the gravitational force is mM Fg G mM 2 Fg G r 2 r

31 27 (9.11 1031 kg)(1.67 1027 kg) 11 2 (9.11 1031kg)(1.67 10 27kg) kg)(1.67 10 kg) (6.67 1011 Nm22 22 (9.11 10 /kg2) 11 2 Nm /kg (6.67 10 N m /kg )) (5.3 1011 m)22 11 (5.3 10 (5.3 10 m) m) 47 3.6 1047 N 3.6 10 N

The magnitude of the electric force is La magnitud de la fuerza elctrica esis The magnitude of the electric force (1.60 1019 C)2 1 ee 19 2 Fe 1 e e (8.99 109 N m2C2) (1.60 1011 C)2 9 2 2 Fe 4p0 r 2 (8.99 10 Nm C ) (5.3 10 2 11 m) 2 4p0 r (5.3 10 m) 8.2 108 N 8 8.2 10 N La relacin entre estas fuerzas es (8.2 8 108 N)/(3.6 47 1047 N) = 2.339 39 The ratio of these forces is (8.2 108 N)(3.6 1047 N) 2.3 1039 . The ratio of these la fuerza elctrica es abrumadoramente mayor que fuerza forces is (8.2 10 N)(3.6 10 N) 2.3 la 10 . 10 . Por consiguiente, Thus the electric force overwhelms the gravitational force. gravitacional. Thus the electric force overwhelms the gravitational force. Since the gravitational force is insignificant compared with the electric force, Dado que la fuerza gravitacional es insignificante en comparacin con la force, Since the gravitational force is insignificant compared with the electric fuerza it can be se puede despreciar. En ese caso, laelectron is then electrn es neglected. The acceleration of the aceleracin del elctrica, neglected. The acceleration of the electron is then it can be8 8.2 108 N F 22 2 a F 8.2 1031N 9.0 1022 m/s2 a m 9.11 1031 kg 9.0 10 m/s m 9.11 10 kg

This is a gigantic acceleration. If it occurred along the electrons motion instead Es una aceleracin acceleration.Si fuera a lo largo del movimiento motion instead This is a gigantic gigantesca. If it occurred along the electrons del electrn, en of centripetally, such an acceleration could boost the electrons velocity close to lugar de ser centrpeta, esa aceleracincould boost the electrons velocity close to la velocidad del electrn of centripetally, such an acceleration podra aumentar 15 one-third of the speed of la velocidad a femtosecond (10 femtosegundo (1015 s)! s)! hasta cerca of the tercio de light in onlyde femtosecond (1015 s)! de un speed of light in only a la luz, slo en un one-thirdCOMMENTS: Note that for the para of the electric force elctrica y fuerza graCOMENTArIOS: Obsrvese que ratio la relacin de fuerza and the gravitational COMMENTS: Note that for the ratio of the electric force and the gravitational vitacional, entre el proton and electron, weobtendra el mismo e inmenso valor de force between the protn y el electrn, se would obtain the same immense value force between the proton and electron, we would obtain the same immense value 2.3 1039 sea cual sea laseparation between the two particles, since both areson fuer2.3 1039 whatever the separacin entre las dos partculas, porque ambas inverse2.3 1039 whatever the separation between the two particles, since both are inversezas de inverso Also notice that for the given tomarse en cuenta quethe electric force square forces. del cuadrado. Tambin debe atomic-scale distance, para la distancia square forces. Also notice that for the given atomic-scale distance, the electric force indicada, a escala atmica, la fuerza elctrica tiene un valor medible, whereas the has a measurable value, the same as weighing an 8-microgram mass, que es como weighing has a measurable value, the same asmientras que la fuerza gravitacional est the pesar una masa deis far below the current limitsan 8-microgram mass, whereas muy gravitational force 8 microgramos, of detection (the highest sensitivity gravitational force is far below the currentmxima sensibilidad a la que se sensitivity 20 detection (the highest ha llegado por debajo de los lmites de deteccin; la limits of N). attained by a measurement of force is near 1020 attained by measurement of force is near 20 al medir unaafuerza es aproximadamente 1010 N.N).

E XAMPLE 2 EJEMPLO 2 2 E XAMPLE

How much negative charge and how much positive charge are Cunta carga negativa y cunta how much positive en los elecHow much negative charge and carga positiva hay charge are there on the electrons and the protons in a cup of water (0.25 kg)? tronesonlos protones de un vaso de agua (0.25 kg)? (0.25 kg)? there y the electrons and the protons in a cup of water SOLUTION: The molecular mass of water is 18 g; hence, 250 g of water amounts SOLUCIN: The molecular mass of wateres 18 g; por consiguiente, 250 amounts SOLUTION: La masa molecular del agua is 18 g; hence, 250 g of water g de agua 23 to 25018 moles. moles.mole has 6.02 1023 molecules, giving (25018) Each equivalen a 250/18 Each mole has 6.02 1023 molecules, giving (25018) to 25018 moles. 23 Cada mol tiene 6.02 10 molculas, que resultan en 23 6.0 10 molecules in the cup. Each molecule consists of two hydrogen atoms (one (250/18) 23 molecules in the cup. Each molecule consists of two est formada por dos 6.0 10 6.0 10 molculas en el vaso. Cada molcula hydrogen atoms (one electron apiece) and one oxygen atom (eight uno) y un tomo there are 10 (con ocho electrons). Thus, de oxgeno electrons tomos de hidrgeno (con un electrn(eight electrons). Thus, there are 10 electrons electron apiece) and one oxygen atom cada in each molecule, and the total negative charge on all the electrons together is electrones). Entonces, thecada molcula charge on all the electrons togethernegativa in each molecule, and en total negative hay 10 electrones, y la carga total is de todos los electrones es (250 g)(1 mole>18 g)(6.02 1023 molecules/mole) (250 g)(1 mole>18 g)(6.02 23 23 molecules/mole) 10 (250 g)(1 mol/18 g)(6.02 10 molculas/mol) 19 (10 electrons/molecule)(1.60 10 C/electron) (10 electrons/molecule)(1.60 10 19 C/electron) (10 electrones/molcula)(1.60 1019 C/electrn)

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= 1.3 107 C 1.3 107 C La carga positiva de los protones es esta misma, pero con signo positivo. The positive charge on the protons is the opposite of this.

EJEMPLO 3 3 EXAMPLE

What is the magnitude la fuerza de atraccin que ejercen los Cul es la magnitud de of the attractive force exerted by the electrons en un vaso water on the los protones second cup de electrones in a cup of de agua sobre protons in a en otro vaso of water at est a 10 of 10 m? agua que a distancem de distancia?

SOLUCIN: According to theel ejemplo example, la carga de on the electrons in SOLUTION: De acuerdo con preceding anterior, the charge los electrones en el 7 vaso cup is 1.3107 107yC and the charge on the es +1.3 1.3 107 considera the es 1.3 C, la carga de los protones protons is 10 C. Si se C. If we que las dos of these charges as pointfuerza sobre los protones es treat both son cargas puntuales, la charges, the force on the protons is

qq 1 qq F 1 F 4p 22 4p00 rr7 7 (1.3 107 C)(1.3 107 C) (8.99 1099N m2/C2) (1.3 10 C)(1.3 10 C) (8.99 10 Nm2/C2) (10 m)2 (10 m)2

1.5 1022 N 1.5 1022 N This is approximately the weight of a billion billion tons! This enormous attracEs aproximadamente el peso de 1018 toneladas! Esta enorme fuerza de atraccin tive los on the protons is precisely canceled by fuerza de large repulsive force sobreforceprotones se anula exactamente por una an equally repulsin igualmente exerted by ejercen los protones on the protons in the other cup. Thus, the cups grande, quethe protons in one cupde un vaso sobre los protones del otro. As, los exert no net forces on neta entre vasos no ejercen fuerzaeach other. ellos.

Consider two toner particles separated by 1.2 105 m; each EXAMPLE 4 Dos partculas de tner estn separadas por 1.2 105 m; cada EJEMPLO 4 of the two particles has a negative charge of 3.0 1014 C. una tiene una carga negativa de 3.0 1014 C. Cul es la What is the electric force that one particle exerts on the other? Treat the toner fuerza elctrica que ejercen las dos partculas entre s? Considrese que las partcuparticles approximately as point particles. las de tner son, aproximadamente, partculas puntuales. force puntuales, la fuerza que una partcula cargasthat one particle exerts on the other is ejerce sobre la otra es F 1 qq 4p0 r 2 (3.0 1014 C)2 (1.2 105 m)2SOLUTION: Como se puede considerar que las partculas as aproximadamente SOLUCIN: Since we may treat the particles approximatelyson point charges, the


(8.99 109 N m2/C2) 5.6 108 N

El signo positivo nos recuerdathat thefuerzais repulsive, tending to pusha alejar diThe positive sign reminds us que la force es de repulsin, que tiende each parrectamente a las partculas entre s, a lo largo de la lnea que las une (vase la figura ticle directly away from the other, along the line joining them (see Fig. 22.4b). 22.4b). Esta repulsin mutua ayuda a mantener dispersas las partculas de tner, y This mutual repulsion helps to keep the toner particles dispersed, so they do not esto evita que se aglomeren en una regin. clump up in one region.

EJEMPLO 5 5 EXAMPLE

Unsimple electroscope for the detectionmedirmeasurement of A electroscopio simple para detectar y and cargas elctricas se compone de consists of two small foil-covered cork ballscon electric charge dos pequeas esferas de corcho recubiertas of lmina 104 kg each suspended1.5 threads 10 cm long colgadas de un hilo de equal metlica; cada una pesa by 104 kg, y estn (see Fig. 22.5). When 10 cm 1.5 de longitud (vase la figura 22.5). Cuando se agregan cargas elctricas iguales a las electric charges are placed on the balls, the electric repulsive force pushes them esferas, la fuerza de repulsin elctrica las aleja, y el ngulo entre los hilos indica la apart, and the angle between the threads indicates the magnitude of the electric

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CAPTULO22 Electric Force and Electric Charge CHAPTER 22 Fuerza elctrica y carga elctrica CHAPTER 22 Electric Force and Electric Charge CHAPTER 22 22 Electric Force and Electric Charge CHAPTER Electric Force and Electric Charge

La fuerza elctrica separa las esferas

haciendo que los hilos formen este ngulo en el equilibrio.

l

30

30

l

rb)

Se descompone la tensin.

T sen 30 T 30 T cos 30

FEn el equilibrio, las componentes verticales de las fuerzas suman cero, al igual que las componentes horizontales.

w

charge. If the equilibrium angle between the threads is 60, what is the magnitude magnitud equilibrium angle between the threads is 60, 60, what ismagnitudecul charge.If de equilibrium angle el ngulo de equilibrio entre los hilos es 60, If the charge.thethe thela carga elctrica. Sibetweenthe threads is 60, what is the magnitude charge.charge?equilibriumangle between the threads is what is the the magnitude of If es la magnitud de la carga? ofthe charge? the charge? of the charge? of SOLUTION: Figure 22.5b shows a free-body diagram for one of the balls. The elecSOLUCIN: Figure 22.5b shows a free-body diagram for one of libre de una de las La figura 22.5b muestra un diagrama de cuerpo the balls. SOLUTION: Figure 22.5b shows a free-body diagram for of the the balls. The elecSOLUTION: Figure along the line joining the two charges and is thus horizontal. In elecelecSOLUTION:acts 22.5b shows a free-body diagram for one one of balls. TheThe equitric force Ffuerza elctrica F acta a lo largo de la lnea que une las dos cargas, por esferas. La acts along the line joining the two charges and is thus horizontal. In equitric force F tric tric force F acts along line line joining twotwo charges and is thus horizontal. In equiforce F acts along the the of the electric repulsion F, the weight w, andIn equijoining the the charges and is thus horizontal. the tension librium, the vector sum equilibrio, lo que es the vector sum of the electricsuma vectorial deweight w, and the tension En la repulsin librium, horizontal.of of electric la repulsionF, weight w, w, theelctrica F, el librium, the the vector sumthe the electric repulsionthe the weightandand the the tenF, F, the librium, vector sumT del zero.debe repulsionEn consecuencia, el componente horiT of the thread must be hilo Accordingly, the horizontal componenttension of tension peso w y la tensin be zero. Accordingly, the horizontal component of the tenser cero. of the thread mustbe zero. Accordingly, the T ofTof the thread mustzero.equilibrar la repulsinhorizontal ycomponent of the tenmust bedebe Accordingly, the horizontal component of of the Tthe thread tensinthe electric repulsion, and the vertical componentthe ten- tension must balance zontal de labalance the electric repulsion, and the vertical component of the tenelctrica, el componente vertical sion must sionsion must balance equilibrar el repulsion, and the vertical componentthe the tenmust balance the electric repulsion, de la tensin debe the electric peso: and the vertical component of of tension must balance the weight: sion must balance weight: sionsion must balance the weight: must balance the the weight: F T sen 30 F F T sen 30 sen sen FT T 30 30 mg T cos 30 mg cos cos mg mgT T cos 30 T 30 30 We can eliminate the tension las ecuaciones, determinando laratio of these equathe problem by taking the relacin entre ellas; Se puedeeliminate la tensin defrom the problem by taking the ratio of these equaWe can eliminar the tension from We We can eliminate the tension from the problemtaking the the ratio of these equacan eliminate the tension from the problem by by taking ratio of these equations, yielding el resultado es tions, yielding tions, yielding tions, yielding F mg tan 30 mg tan F F tan tan 30 Fmg mg 30 30 From Fig. 22.5a we see that the distance between the balls is r 2l sin 30, so From 22.5a22.5 we see the the distance between the is r 2l sin so En la figura22.5asee see that the la distancia entre las balls ises2l 2l 30,30, so From Fig.Fig. 22.5a tellsobserva que distance between the esferasrr sin 2l sen 30, y we se us From Fig. Law we that that distance between the ballsballs is r = sin 30, so Coulombs acuerdo con la ley de Coulomb, entonces, deLawtells us CoulombsLaw tells us Coulombs Law tells us Coulombs q2 1 q22 1 F 1 q2 q2 1 30)2 F F 4p0 (2l sen 2 22 F 4p (2l sen 30) 4p4p00 sen 30) (2l (2l sen 30) 2 0 0 Equating these two expressions for F, we find Equating Equating these twotwo expressions F, we findfind Equating these two expressions for F, we find Se igualanthese expressions for for F, we resulta las dos ecuaciones anteriores, y q2 1 q22 1 q2 q2 mg tan 30 1 mg tan 30 1 30)2 mg tan tan 30 4p0 (2l sen 2 22 mg 30 4p (2l sen 30) 4p4p00 sen 30) (2l (2l sen 30) 2 0 0 and and andpor lo que and q 14p mg tan 30 2l sen 30 q 14p 0 30 2l 30 30 14p tan tan 30 2l sen q q 14p0mg00mg tan 302l sen sen 30 mg 0 12 2 2 4 m/s 2(4p)(8.85 1012 C2/Nm2)(1.5 10 4 kg)(9.81 2 2)(tan 30) 2(4p)(8.851012 12/Nm2)(1.5 10 4 kg)(9.81 m/sm/s22)(tan 30) 2(4p)(8.85 10 12 C2/Nm2)(1.5 10 4 kg)(9.81 m/s2)(tan 30) 10C2 C2 /Nm2 )(1.5 104 kg)(9.81 )(tan 30) 2(4p)(8.85 (2)(0.10 m)(sen 30) (2)(0.10 m)(sen (2)(0.10 m)(sen 30)30) (2)(0.10 m)(sen 30) 8 3.1 108 C 3.1108 C C 3.1 108 C 108 3.1

FIGURE 22.5 (a) Two equal charged FIGURE (a) equal charged FIGUrA 22.5 (a) Two esferascharged a) Two equal de cargas FIGURE 22.5 (a) Twothreads. (b)charged FIGURE 22.5 by Dos equal Free-body balls suspended balls suspended de hilos. b) (b) Free-body iguales colgadas by threads.(b) Free-body ballsballs suspended by threads.Free-body de suspended by threads. (b) Diagrama diagram for the right ball. diagram for the ball. ball. cuerpo libre para la esfera diagram for the rightrightball. derecha. diagram for the right

can you conclude about the signs of charges? canpuede decirse acerca the los signos of these cargas? you conclude about de signs of de thesecharges? can you conclude about the signsthese charges? esas QUESTION 2: Suppose that the electric force between two charges separated by a disQUESTION 2: Suppose4 electric force between two two charges separated by a disthat the electric force between charges separated QUESTION 2: Suppose thatthat the electric elctrica entre two cargas separadas por a disQUESTION 2: Supngase N. la fuerza be the electric dos charges separated distance PrEGUNTA m is 1 104 queWhat will force between force if we increaseby aby 1 m es tance of 1 2: Suppose the the dis4 m is 1 10 tanceof 1 N. is 1104 N. laN.What willdistancia aumenta a 10we increase the distance of 1m Cul 104What will willbe the electric force if increaseathe distance tance 10m is 1 ser N. What be the electric force if we we increase the distance 1toof 1 m? To 100 m? 4 fuerza si la be the electric force if m?, y 100 m? tance 10 to 10 m? To to 10 m? To 100100 m? to 10 m? To 100 m? PrEGUNTA 3: m? esferas separadas por cierta distancia, tienen cargas elctricas iguales, Dos QUESTION 3: Two balls, separated by some distance, carry equal electric charges and 3: balls, separated by some distance, carry equal electric yQUESTION 3: Two balls,separatedsome other. If we transfer a fraction ofcharges de la ejercenrepulsive electric separatedeach distance, carry se transfiere charges and and entre ellas una force by by some distance, carry equal electric charges QUESTION 3: TwoTwo balls,fuerza elctrica de repulsin. Si equal electricuna fraccin and QUESTION exert a on the electric exert a repulsive electric force eacheach other. If transfer a fraction of elctrica? a repulsive electric force on each other. If we transfer a fractionthe the electric on aumentar o we transfer la fraction of electric carga elctrica de una a otra esfera disminuir a fuerza of exert a repulsive electric force on will other. If we force increase or decrease?the electric exert of one ball to the other, charge the electric charge of 4: to partculas estn the electric force increase or decrease? charge of oneone ballthe the other, will the electric force increasedecrease? PrEGUNTA one Dosto other, will will electric force increase or or una ejerce una fuerza charge of ball ball to the other, the a 3.0 m una de otra; cada decrease? QUESTION 4: Two particles are separated by a distance of 3.0 m; each exerts an elecQUESTION 4: N particles are separated by a distance of 3.0 m; carga elctrica exerts an elecelctrica de4: Two particles separated bypartcula tiene 10 m; m; la exerts an an elecSi una distance of QUESTION 4: TwoTwosobre areotra.separatedaby a distance3.0 3.0 eacheach exertselec- que QUESTION 1.0 particles la are If one particle carries 10of veces much electric charge tric force of 1.0 N on the other. times as each tricforce of1.0 N onother. If one particle the other. If one carga carries tric tieneforceof N on the la magnitud onelaparticlemenor?10 times as much electric charge force la otra, cul es the other. If de particle carries 10 times as much electric charge tricthe other,1.0N on the magnitude of thecarries 10 times as much electric charge as of 1.0 what is smaller charge? asthe other, what is the magnitudethe the smaller charge? the other, what is magnitude of of smaller charge? the magnitude of the smaller charge? as the other, what is the (B) 10 C as (A) 10 pC (C) 10 C 10 C (D) 10(D) 10 kC kC (A) 10 pC (B) 10 C (C) (A) 10 (B) 10 (C) 10 (D)10 kC (A) (A)pC pC 10 10 pC (B) (B)C C 10 10 C (C) (C)C C 10 10 C (D)(D)kC kC 10 10

QUESTION 1: Suppose that the electric force between two charges is attractive. What QUESTION 1: Suppose that the electric force between charges is attractive. What charges is attractive. What QUESTION 1: Suppose thatthat the electricentre dos cargastwocharges isde atraccin. Qu QUESTION 1: Suppose the signs of these charges? PrEGUNTA 1: Supngase theelectric force between twotwo can you conclude about que la fuerza force between elctricas es attractive. What

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2 2 3 H E SU PE RPOSI TI ON 2 2 ..3 TLA SU PE RPOSI TI ON 22.3 THE SUPErPOSICIN O F E EC TRI C FORC E S O F E LLE CTRI C FORC E S

ELCTrICAS

DE FUErZAS

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The electric force, like any other force, has magnitude and direction; that is, the The electric force, like any other force, has aamagnitude and aadirection; that is, the La fuerza elctrica, como cualquier otra fuerza, tiene magnitud y direccin, esto es, la electric force is vector. According to Coulombs Law, the magnitude of the electric electric force is aavector. According to Coulombs Law, the magnitude of the electric fuerza elctrica es un vector. De acuerdo con la ley de Coulomb, la magnitud de la force exerted by a point charge q on point charge is force exerted byque una chargepuntualaq ejerce sobreqis carga puntual q es fuerza elctrica a point carga q on a point charge q otra F F qq 11 qq 4p 2 4p00 rr 2 (22.8) (22.8) (22.8)

La direccin de esta fuerza se da a lo largo de la lnea que va de una carga a la otra. The direction of this force is along the line from one charge to the other. As illustrated The direction of this force is along the line from one charge to the other. As illustrated Como se muestra en la figura 22.6, se puede representar la fuerza por un vector F que in Fig. 22.6, this force can represented by vector F pointing along the line from in Fig. 22.6, this force lnea berepresented by aavectorotra. apunta a lo largo de la can be va de una carga a la F pointing along the line from que onecharge to the other. charge to the other. one Si hay varias cargas puntuales q , q , q , que ejercen al mismo tiempo fuerzas 1 2 3 If several point charges 1, 2, 3, simultaneously exert electric forces on the If several point charges fuerza neta . .simultaneously exert electric forces on the elctricas sobre la carga q, laqq1,qq2,qq3,. .. .sobre q se obtiene calculando la suma vectorial de charge q, individuales (vase is obtained by taking the vector sum the individual a las charge q, then the net force on qqis obtained by taking thelos vectores ofthe individual forces las fuerzasthen the net force on la figura 22.7). As, si vector sum ofque representanforces (see Fig. 22.7). Thus, if the vectors representing the individual F , forces produced (see Fig.individuales producidas por q1, q2, q3, son F1, F2electric forces produced fuerzas 22.7). Thus, if the vectors representing the individual ,electric respectivamente, 3 are F , F ., respectively, then the net force is by 1, 2, 3 by qq1,qq2,qq3fuerza neta,F22,F33,. .. .., respectively, then the net force is entonces la , ,. .. .. .are F11 es , F , FF F F F F11 F22 F33 (22.9) (22.9) (22.9)principio de superposicin Superposition Principle of electric forces Superposition Principle of electric forces de las fuerzas elctricas

La ecuacin (22.9) expresa el principio de superposicin de las fuerzasAccording Equation (22.9) expresses the Superposition Principle of electric forces. elctricas. Equation (22.9) expresses the Superposition Principle of electric forces. According De Eq. (22.9), theforce contributed by each aporta cada carga es of the presence ofla to acuerdo con force contributed by que charge is independent of the presence to Eq. (22.9), theesa ecuacin, la fuerza eachcharge is independentindependiente deof presencia charges. For instance, Por charge qq does not qaffectafecta la interaccin de la the other charges. For cargas. the charge 2 does not 2 no the interaction of charge the other de las demsinstance,the ejemplo,2la carga affect the interaction of charge carga q1q; it merely suma its propia interaccin conq. This simple combination law is with q; it merely adds its own interaction with q. Esta simple ley de combinacin qq1with con q; slo adds su own interaction with q. This sencillacombination law is 1 es otro hecho emprico importante electricde las fuerzas elctricas. As como las fuerzas an important empirical fact about acerca forces. Since the contact forces of everyday an important empirical fact about electric forces. Since the contact forces of everyday de contacto such as the normal force and the friction force, arise normal y la de fricexperience, en la experiencia cotidiana por ejemplo la fuerza from electric forces experience, such as the normal force and the friction force, arise from electric forces cin se deben a fuerzas elctricas entre los tomos, tambin ellas obedecern el prinbetween the atoms, they will likewise obey the Superposition Principle, and they can between the atoms, they will likewise obey the Superposition Principle, and they can cipio de superposicin, y se pueden combinar en una simple suma vectorial. Por cierto: be combined with simple vector addition. Incidentally: The gravitational forces on the be combined with simple vector addition. Incidentally: The gravitational forces on the las fuerzas gravitacionales sobre la Tierra, y dentro del sistema solar, tambin obedecen Earth and within the Solar System also obey the Principle. Thus, all Earth and within the Solar System also las fuerzas SuperpositionPrinciple. Thus, all al principio de superposicin. As, todas obey the Superposition inmediato se apegan en el entorno the forces in ourimmediate environment obey this principle. We have already made immediate environment obey this principle. We have already the forces in ourYa se ha usado mucho el principio de superposicin al estudiarmade a este principio. mecmuch ahora queda claro que la superposicinourstudy of mechanics; como we recognize use of the Superposition Principle in de study of mechanics; now son las fuerzas muchyuse of the Superposition Principle in our las fuerzas mecnicas,now we recognize nica, that the superposition of mechanical forces, fuerzas elctricas. that the superposition of mechanical forces, such as contact forces, hinges on the superposide contacto, se debe a la superposicin de lassuch as contact forces, hinges on the superposition of electric forces. tion of electric forces.La fuerza neta sobre q se obtiene con el mtodo del paralelogramo.

F q rFuerza sobre q debida a q'. Fuerza sobre q debida a q2.

F1 F2

F2 q

F1Fuerza sobre q debida a q1.

q'Estas cargas tienen el mismo signo, y por tanto se repelen.

q1

q2

FIGURE 22.6 A charge q exerts an FIGURE 22.6 A charge q q ejerce una FIGUrA 22.6 Una carga exerts an electric force F on the charge q. electricelctricaon sobre la carga q. fuerza force F F the charge q.

FIGURE Dos Two puntuales q 1 FIGURE 22.7 Two point charges q FIGUrA 22.722.7 cargaspointcharges1qy 1 and 2 fuerzas elctricas F F and qq2exert electric forcesF11yand F22on q2 ejercen las exert electric forcesF 1 and F on 2 the point chargeq. La net force on is the point charge q. Thefuerza netaonqqis sobre la carga puntual q. The net force sobre the vector sum of these forces.fuerzas. qthe la suma vectorial deforces. es vector sum of these esas

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CAPTULO Electric Force and Electric Charge CHAPTER 22 22 Fuerza elctrica y carga elctrica CHAPTERCHAPTER 22 Force and Electric Charge Charge CHAPTER22 Electric Force and Electric Charge 22 Electric Electric Force and Electric

Two point charges Q and Q are separated by a distance distanDos cargas puntuales, +Q y Q, estn separadas por la d, as EXE M P L O 6 E J AMPLE Two point 22.8. AQ and Q are separated by a distancedistance Two point la and Q charge q is equidistant a d, Two point muestraQfigura separated by a estas shown in Fig.charges charges Q and Q are separated byfrom as d, as EXAMPLE 6 cia 6 como charges positive pointareEquidistante a distance d, as y EXAMPLE d, EXAMPLE 6 shown in Fig. 22.8. A positive 22.8. charge q is equidistantcargas shown in Fig. positive point the electric forceis equidistant shownfrom distancia 22.8. A positivemedio. CulqesFla fuerza from in Fig. 22.8. A x de su point charge q is equidistantfrom from these una carga puntual q,x una their midpoint. What is point charge on q? hay charges, at a distance a punto y F1 debido a que Q these charges,charges, at a distancetheir midpoint. What isWhat is the force F on q? F on q? aa distance x from x from their midpoint. the electric force F on q? these at q?distance x from their midpoint. What isthe electric electric force these charges, at elctrica F sobre Q es de repulsin. SOLUTION: As illustrated in Fig. 22.8a, the charge Q produces a repulsive force r d/2 q SOLUTION: As illustrated in Fig. 22.8a, the +Q produce repulsive force x SOLUTION: As illustratedin Q produces charge charge Q produces vector onSOLUTION:q,Asillustrated la Fig. 22.8a, thelachargeQ produces aarepulsive force force the charge and ilustra in Fig. 22.8a, the attractive force. Thus, the a repulsive SOLUCIN: Como the charge figura 22.8a, an carga Q produces una fuerza de x on the chargefromcarga q,chargeQ produces an toward Q. Fromforce.thevector vector on theq, and theand la carga Q2 produce attractiveforce. Thus, the As,the on sin sobre charge q, charge Q produces an unaan attractive the geome- el y the charge Q produces fuerza de Thus, Thus, d/2 Frepulthe charge q, and theandthe vector F points attractive force.atraccin. vector 1 points away la Q, Fofpointssepoints de Q, and the and F thepoints 2toward Q.De to theFrom the geomeawaytheaway fromel vector of Fvector F hacia and Q Q. charge la from +Q, yfrom each these dirige toward toward acuerdogeomeF1 F2 vec1 points1 away distance Q, vector 2 points Q Q. From the con aleja 1 Fig. F tryFtor F122.8a, from Q, and thevector F2 chargespoints Q.From the geome2 Q F1 F2 try Fig.222.8a, the distancela magnitudes of theuna de andQ and the charge try 22.8a, the from each the of individual Q +Q to try of 2x ofla dfigura 22.8a, from each ofde cada the Q andQ to Q Q a qgeometra deFig.>4. distancedistance from the charges chargescargasto theycharge charge is r of Fig. 22.8a,2the Hence, thedistanciaof eachchargesQlas Coulomb forcesthe 2 q is r es r2 Q Hence, Por consiguiente, the individual Coulomb forces laqcarga 2x 2 d 2 2 ared 2>4. the magnitudes of las magnitudes Coulomb forces is is q 2x d >4. the magnitudes of the of the individual Coulomb exertedrbyqQ and2x>4.Hence, Hence, the magnitudesindividual de las fuerzas in- forces b) exerted exerted by Qand Q are by Q and Qare dividualesbyQ andQ que ejercen +Q y Q son de Coulomb are exerted y qQ Los componentes x 1 qQ 1 F1 F2 (22.10) qQ se anulan 1 2qQ 14p1 2 qQ qQ qQ qQ (magnitudes) 12 4p01 F1 F2 F2 r 2 01x d 4 (22.10) (22.10) (magnitudes) (magnitudes) F1 FF1 4p 4p 22 (22.10) (magnitudes) (22.10) 2 x 4p 22 2 2 2 2 4p0 rr4p0 4p0 x d x4 d 4 0 r 0 0 4 d F2 sen F1 sen From Fig. 22.8b we see that in the vector sum F F1 F2, the horizontal comx From (xFrom Fig.se see thatin thatvector and F F1 F F1the F2, se anulan los From Fig. 22.8b 22.8b we queFen in suma vectorial F2,1 horizontal comEn la figura 22.8b we observasee the cancel, sum thesum1F =components (yhorizontal ponents Fig. 22.8b we see thatandthe vector sum F verticalF2Fthe horizontal com- comcomponents) of F1 in 2 la the vector F , + F2 the com ponentsadd,components)of(componentes x)2andFtheand,thequeeach componentes(y com(x components) ofF1 andcomponentde theyvertical as los components F2 F1 componentes giving a net vertical F2 and F and 1 verticalvertical individual ponents (x horizontales F1 and cancel, twice large components (y components) ponents (x components) ofF21cancel, cancel,asF2 y components (y components)(componentesgivingsuman,component unhas a large as each as each F2 cos F1 cos verticalesponents) Thus y) se verticalthe net componentasylarge as vertical que xindividual ponents) add, giving in vertical y se obtiene twice add, giving net a net component F componente each individual vertical component.add,aa netthis case vertical forcetwice astwice as large but no es el component individual vertical component.of the in this case thein Fig. 22.8b, theF component buttiene doble de vertical component. Thuscase thecase force F has aayycomponent but F xbut no x componente in this individual. force F caso, fuerza neta of no F no vertical cada terms Thus angle in this net the net force lahas a y component x component. Thus vertical shown net En este has component. In 1 component. In terms oftermsEn funcin F cos 22.8b, the componenttheF1 un componente y, y nocomponentthe F isdelin Fig. . Sincetheyycomponent 22.8b, of F1 uno x. angle shown shown 22.8b, 22.8b,are figura of ngulo Fig. these la y component component.yInofthe angleofangle in Fig. in que muestratheequal, of F1 of component. In terms the shown 2 is F1 cos , and the 2 el componente1and theand F1y cos of F componente Since.theseare equal, En es are cos equal, the isforce cos , ycos , es the component F is F2 cos F2 Since these arethe y los componentes y F is F de F component of 22 is F2 cos . de cos , netisF11is thenand the1yycomponent, yFel is of F22 cos . ySincetheseF2 equal,.the se suman. vista force is son iguales, la fuerza neta es, entonces: net de que then is then net then net force is force F Fy F1 cos u F2 cos u 2F1 cos u FIGURE 22.8 (a) The charges Q and F Fy F Fy F1 cos u uF2 cos u 1 cos u 1 cos u FIGUrA FIGURE(a)The charges QQ Las cargas +Q y q. FIGURE 22.8 (a) The charges Qand 22.8 and F2 (a) The charges F cos u F cos 2F1 2F F Fy F11cos u F22cos u 2F cos u Q exert forces F1a) 22.8 on the charge andQ and FIGURE 22.8 (22.11) qQ 1 ejercenexertfuerzasFvector F2 onF thechargeq.q. lasQ exert1forcesFsobre la carga q.charge q. F1 and 2 F ontheF charge y Fsumand F . the Q exert forces The net forceforces F1and 21 1 2 on is the (22.11) (22.11) Q 2 u 2 (22.11) (22.11) 1 2 qQ cosqQ qQ 12>4 La fuerza neta netla suma vectorial FF.2. . 4p0 2 1x d 2 cos u cos u (b)The net forcecomponentsthe F1 F11 + 2F2. F2. x and y es the vectorsum and 1F2 F The net The is the vectorof vector sum F1 force is force is sum F 2 4p222 cos 2u 2 2 b) Los componentes x y yycomponents of F and F . de y 2. 4p0 x 4pd >4 d >4 0 x d x (b) The xxand yycomponents F1 F1FandF2.1 0 >4 (b) The (b) The componentsof F1and F2. and x and of 2 1>2 From Fig. 22.8a, we see that cos u 1 d>(x2 2d 2>4)2 1>2 therefore , and 2 1 2 De acuerdo 22.8a, we see 22.8a, cos =1d>(x22 +dd2>4)1>2;,and 1>2, and therefore From Fig. con la figura that see that cosd>(x 1 d>(x2/4) d por consiguiente, From Fig.we see thatcos u 22 u 2 d >4) , 2andtherefore 22.8a, we cos u d/(x >4) therefore From Fig. 22.8a, qQd 1 (22.12) F Fy qQd (22.12) 2 2 4p1 1 qQd3>2 qQd (22.12) (22.12) F Fy F Fy01 (x 2 d >4) 3>2 (22.12) F Fy 4p 3>2 2 d 22 2 3>2 2 4p0 (x4p0 d(x>4) d >4) 0 (x >4) a una gran distancia de COMENTArIOS: that if the charge q is at a la carga q est COMMENTS: Note Debe considerarse que si large distance from the two charges 2 las dos cargas Q, entoncesthecharge withat 2alarge2distancedistance from ,xy .ascharges COMMENTS: Note that if compared despreciar d a large from thecon charges COMMENTS: Note se charge is x large en comparacin 32 charges COMMENTS: Note that if that if the charge so at d 24)32 (x )twox 3 Q, then2d 2 can be neglected the puede qqis ata, q is(x 2 distancefrom the2two the two 2 3/2 22 (x2)3/2 = x3. En ese caso, la fuerza F22 proporcional a21/x3; esto 22 2 2 d 22 2 32 (x2)32 x 33 2 3232 3 (xQ,then d can be neglected comparedwith x ,with x esso 4) 32 4) + d Q, can d can be neglected compared so (x 4) Q, then is then proportional to 1x3; with ,so (x ,decreases in proportion2)32 (x The force /4)d thenbe neglected comparedthat is,xthe force d(x d (x ) x .. x . F 33 es, la fuerzaFdisminuye en proporcin1xinversois, thatforce decreases in proportion The is then proportional to al that 3 the is, de force decreases in proportion The force force F is then proportional to 1x del cubo the la distancia. As, aunto The inverseFcube of proportional to 1x althoughthe force decreases in proportion the force is then the distance. Thus, ;;that is,; the force contributed by each que the fuerza aportada cubedistance.Thus, although the una fuerza del inversoeachby each la inverse cube of the distance. Thus, although es force contributed by each to charge the cube of por cada carga Q puntual the force contributed by del to Q is an inverse-square force, the net although different behavior, to point the inverse inverse the of the distance. Thus,force has athe force contributed cuadrado, la fuerzais an inverse-square force, the net force has a different behavior, neta tiene un comportamiento diferente, porque a grandes dispoint charge Q is an inverse-square force, the net the net force has cancel the behavior, point atpoint Q Q is an inverse-square by oneforce has a different behavior, charge charge the force contributed force, charge tends to a different because large distances tancias la at largeque aportathe force contributed by one charge tends to otra. En el fuerza distances una carga tiende a anular la aportada por la cancel the because because at large distances thefurther discuss thisby one charge tends to cancel the because at large distances the force force contributed 1x3 behavior cancel force contributed by the other. We willcontributed by one charge3tends tofar fromthe captulo 23 se continuar other. We will further discuss this 1x 1/x3 a grandes disdescribiendo este comportamiento de 3 force contributedoppositeother. We will further further discuss this 1x3 behavior far from force force equal contributed by the behavior far from a tancias contributedbycargasby the other. We will discuss this 1x behavior far from pair of de un and de the charges y opuestas. 23. in Chapter par opposite charges in Chapter 23. aapair ofaequalof equal and iguales charges in Chapter 23. pair and opposite pair of equaland opposite charges in Chapter 23. If an arrangement of de cargas puntuales es simtrica de way, themanera, conof Si una distribucin point charges is symmetric in some alguna calculation freIfforcearrangement of point thefuerza neta. is symmetric anterior, calculation de If an arrangement point is symmetric the position the calculation the netIfan arrangement clculo Incharges charges En el in somein some the charge q, of cuencia an simplifica el of pointofchargesis symmetric ejemplo way, theway, the calculation of se is often simplified. de la previous example,in some way, of la posicin of the netforce isoften is often simplified.previous indio como positionpositionchargeq, lathenet qforceisthe charges Qcargastheprevious example, theresultado ofcomponent de carga the net force simplified. In the resulted y a cancellation of one the charge q, es equidistante a las and Q, In the previous example, the la anulacin charge q, equidistant from often simplified. In +Q y Q, example, the position of the of the equidistant from the Q distribucin the in cancellation of one of as component un componenteadethecharges charges Q, resulted resulted in a cancellationcomponent equidistant from la symmetric and Q, resulted result can be even simpler,onein of equidistant frommorefuerza. Si laandQ and Q,msaacancellation resultado se puede the force. For the chargesQ arrangement,es in simtrica, el of one component offollowing example.For symmetric ejemplo arrangement, thecanbe even be even simpler, as in the force. For aams, como en symmetric siguiente.result can be evensimpler, as in of the force. simplificar todava more symmetric arrangement, the theof the force. For more a more el arrangement, the result result can simpler, as in the following example. the following example. the following example. Como en la foto que abri el captulo, la figura 22.9a es una miEJEMPLO 7 Similar to the chapter photo, Fig. 22.9a is a scanning electron crografa to the chapter barridoFig.partculasade tner. Se ve que de EX AMPLE 7 Similar electrnicathe photo, de 22.9a is 22.9a is a scanning electron to the to photo, photo, Fig. ascanning electron is scanning electron EXAMPLE 7 lasSimilarSimilarchapterchapter Fig. 22.9a que se encuentra of el EXAMPLE partculas toner particles. We aquella 7 laEXAMPLE 7 micrograph of of tner es casi comosee that the arrangement enof distribucin de micrographde toner particles. We see that the arrangement micrograph of toner particles. We see that the arrangement of micrographone tonerthe center of a hexagon the arrangement of of is at particles. We see that and the other six toner particleshexgono, y las otras seis estn en los vrtices del mismo. Si las partcucentro de un is nearly such that toner particlesparticles is nearly Ifonethat one is at theof hexagonand the and the other six toner is nearly such that theis at the center center of a and theother aretonertner tienen the hexagon. suchsetoner particles of aahexagonhexagonand aresix lasat the vertices is nearly such that one is at the center carry equal charges other los de particles of cargas iguales y distribuyen exactamente en el centro y six are at the vertices verticeshexagon.If the ofIf hexagon, what is equal charges charges and are are at the of center the hexagon. a particles carry equal equal and of are at de verticesof the hexagon.fuerza toner particles carry the netchargesand are arranged theun hexgono cul and vertices neta the toner particles carryforce on theare vrtices precisely at the the es la If the toner sobre la carga central? Para simplificar, arranged preciselyprecisely at theandvertices vertices of a hexagon, what isforce onthe arranged at for center center eachcarga puntual a point net the net arranged preciselyat the simplicity vertices ofaahexagon, asubicada charge locatedthe hexagon, what is the net centro. central charge? Assumethe center and thatand ofparticle acts what is theen suforceonforce on the supngase que cada partcula equivale a una central charge? AssumeAssume for simplicity that each acts as aaacts ascharge located located central Assume for simplicity that each particle acts as point a point charge central charge?charge? for simplicity that each particle particle point charge located at the center of the particle. at the center of the particle.particle. at the center of the at the center of the particle.


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22.3 La superposicin de fuerzas elctricas

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a)

b) Las partculas de tner forman un hexgono que rodea a una partcula central.

Estas dos partculas ejercen fuerzas opuestas. Q1 F4 Q6 F3 F2 Q5 Q2 F5 F6 F1 Q4 Q3

b)

Estas dos partculas ejercen fuerzas opuestas. Q1 Q2

Cada contribucin de fuerza de repulsin se aleja directamente de la carga que la ejerce.

FIGUrA 22.9 a) Micrografa electrnica de barrido donde se ven partculas de tner para impresora lser. b) Fuerzas elctricas sobre la partcula central, debidas a cada una de las otras seis.

SOLUCIN: Las cargas iguales se repelen, por lo que la fuerza sobre la partcula F4 F central, debida a cualquiera de las dems cargas,5se alejar directamente de esa carga central. Ya que todas las cargas son iguales, y la distancia del centro a cada vrtice es Q6 F3 igual, cada una de las fuerzas aportadas por las otras Q3 cargas tendr la misma seis F6 magnitud. La figura 22.9b muestra los vectores fuerza elctrica sobre la partcula F 2 central, debidos a cada una de las seisFpartculas 1de tner que la rodean. Al examinar este diagrama se observa que los vectores fuerza se anulan por pares; por ejemplo, Q5 Q4 F1 balancea exactamente a F4. La fuerza neta sobre la partcula central es cero. Cada mutua de de partculas Por lo anterior, la repulsincontribucin lasfuerza de no slo las mantiene disperrepulsin se aleja directamente sas, como se mencion ende laejemplo 4, ejerce.que tambin tiende a mantenerlas en el carga que la sino equilibrio. Las fuerzas de repulsin entre partculas, tomos u otras entidades, producen con frecuencia estructuras hexagonales estables, como los ejemplos que se ven en la figura 22.10. a) b) c) d)

FIGUrA 22.10 Debido a la repulsin mutua, muchas clases de objetos tienden a formar distribuciones hexagonales: a) bolas de billar, b) perlas microscpicas de poliestireno, c) vrtices superconductores y d) tomos en un cristal de xido.

revisin 22.3

PrEGUNTA 1: En el ejemplo 6, supngase que en lugar de las cargas Q, las dos cargas Q son positivas. En ese caso, cul es la direccin de la fuerza elctrica F sobre la carga positiva q?

PrEGUNTA 2: Tres cargas puntuales idnticas estn en los vrtices de un tringulo equi-

ltero. Una cuarta carga puntual, idntica a ellas, est en el punto medio de un lado del tringulo. Como resultado de las tres contribuciones a la fuerza elctrica, debidas a las cargas en los vrtices, la cuarta carga: (A) Est en equilibrio y permanece en reposo (B) Es atrada hacia el centro del tringulo (C) Es impulsada hacia fuera del tringulo

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CAPTULO 22


CHAPTER 22 Electric Force and Electric Charge

CHAPTER 22 Electric Force and Electric Charge

S PErPOSICIN UErZAS PR OB LEM - SOLVINGUT ECHNIQUES D E FSUPERPOSITION OF ELECTRIC FOR T C N I C A TE A H NIQ O L U DE PrOBLEMAS RO BL EM- S O LVI NG S P C r A r E SU E S C I NSUPERPOSITION OF ELECTRIC FORCES EL CTrICAS

charge exerting the force and the charge ac To find the total electric force on one charge due to an Para calcular la fuerza elctrica total sobre una carga, la fuerza y la carga sobre la que acta. Se puede facilitar charge of several the charge acted upon. It may be easiest to put the tail of each contrib To find the total electric force on one charge due to an arrangement exerting the force and charges (such as the debida a una distribucin de varias otras cargas puntuales other pointponer la cola de cada vector fuerza individual en mucho contributing vector arrangement of several otherlas cargas Q del ejemplo 6), se debe calcularof Example 6), you need la que actathe point charges (such as the point may be easiest to put the tail of eachto calculate (como enat the charge acted upon (as in Example 6 charges Q la (como la carga sobre el ejemplo 6) y aleat the charge in Example 6), pointing away from the charge point charges Q of Example 6), you need tofuerzas elctricas individuales. Laacted upon (asde la cargaThe ejerce la fuerza, cuando las cargas exerting the force for c calculate the vector sum of the individual electric forces. que magjndose suma vectorial de las vector sum of the individual electricfuerza elctrica sobre unaawaythe electric forceexerting the charge q ydue to with like signs, que ejerce the charge exerting t of from the on a signos iguales, dirigindose a la carga and toward magnitud de la forces. The mag- nitudecarga puntual charge son de point force for charges like signs, point nitude of the electricq, producidapoint charge q due to each individual and toward the charge exerting diferentes.for unlike signs. force on a por cada carga puntual individual q, se de- charge q is given by Eq. 22.7, la fuerza, para signos the force each individual point chargecon is given by Eq. 22.7, =F )qq/r)2;qqr 2; and the direction of this electric q la ecuacin (22.7), F (1/4 unlike signs.la (140 y termina 0 sense propiedades geomtricas Use geometry to decompose each vector int para descomponer cada F (140 ) qqr 2; and thede esta fuerzathis electrichacia qis toward q if q and q have unlike signs, and away direccin direction of elctrica es force Use geometry to decompose each vector into its x and y (and, if necessary, z) components. You can si q y q tienen vector en sus componentes x y y (y z, si es necesario). force is toward q if q signos have unlikeysigns, and away from q if q and q have like signs. The techniques for and q diferentes, se aleja de q si q y q tienen signos (and, if necessary, z) Entonces se puedecan thenel vector the total electrictotal vector by separately su components. You obtener obtain fuerza elctrica force from q if q and q have like Las tcnicastechniques for suma vectorial de los sum of the electric force vectors of iguales. signs. The para evaluar la evaluating the electric force vector by separately summing the the total vector x, y, and y sumando por separado los componentes x, z y z de los evaluating the vector vectoresthe electric forceson las mismas quechargessuma same as for the sum of any other sum of fuerza elctrica vectors of several para la are the x, y, and z You can de la fuerza elctrivectores individuales. La magnitud vectorial for the sum of any several charges are the same asde cualquier clase. other kind of vector. ca total se obtiene de la forma acostumbrada, con F 2F 2 F 2 F 2 x y z kind of vector. Recurdense las tcnicas bsicas para calcular sumas F 2F 2 F 2 F 2 . En sums: ejemplos sobre clculos Recall the basic techniques for x calculating vector muchos y z de for the forces on a given vectoriales: primero, vector con First, un diagrama x, y Recall the basic techniques for calculating dibujarsums:cuidadodraw a careful diagramfuerzas elctricas, algunos componentes de la fuerza para las fuerzas sobre determinadacharge q1, due to several otheranularn. Si ,se seleccionanz axes judiciously, you can often achieve cancellation carga q debidas a se charges q q , . . y, and los ejes x, y y z con cuidado, x, ., with First, draw a careful diagram for the forces on a given 1 2 3 varias otras cargas q2 q . . ., with each electric force can often frecuencia throughlograr la anulacin except one judiciously, you charge q1 due to several other charges q2,, q3,, y cada vector fuerza elctricavector con achieve linese puedeofthe components de todos los(see Example 6). along the cancellation all 3 est a along the la lnea que pasa por componentes, excepto uno (vase el ejemplo 6). ponents ejerce each electric force vector lo largo de line through the la carga que except one (see Example 6).

Not fuerzas elctricas protons exert electric forces No slo los electrones y protones ejercenonly electrons andentre s, sino tambin mu- on each other, b Not only electrons and protons exert electric magnitudes de esas fuerzas elctricas se calculan conelectric forces are given by E other particles. The magnitudes of these la ecuachas otras partculas. Las forces on each other, but so do many other particles. The magnitudes of these electric forces are given by Eq. (22.7) with En la tabla 22.3 se 22.3 lists the elec cin (22.7), usando los valores dethe appropriate values of the electric charges. Table carga elctrica adecuados. the appropriate values of the electric charges. Tableelctricas de algunas partculas;of el captulo 41 se ver in Chapter 41. An lists particles; a more complete list will be found presenta una lista de las cargas 22.3somethe electric charges en some particles; a more complete list will be foundantipartculas tienen cargas are opposite to those of the a electric Antiparticles have una lista ms completa. Las in Chapter 41.charges that elctricas que son opuestas corresponding parti electric charges that las de sus correspondientes partculas; por ejemplo, un for (or positron) has charge e, the antiproton has are opposite to those of the corresponding particles; antielectrn (o positrn) tiene la ple, the antielectron example, the antielectron carga +e, el antiprotn tiene carga e, el antineutrn tiene carga 0,so on. (or positron) has charge e, the antiproton has has charge 0, and etctera. antineutron charge e, the antineutron has charge 0, and las on. Todas so partculas conocidas tienen cargas que son mltiplos enteros de la carga funda- integer multiple of All the known particles have charges that are some mental; esto es, las cargas some integer multiplethat is, the charges All the known particles have charges that aresiempre son 0, e, 2e,the fundamen- arede que no existan 3e, etc. Why no tal charge; of 3e, etc. La causa always 0, e, 2e, tal charge; that is, theotras cargas es un misterio, para3e,exist Why no otherno tiene classical physics offers no explanation. Sin charges are always 0, e, 2e, el cual lais a mystery forcharges explicacin. Dado que etc. fsica clsica which cuantizacin de is acarga la mystery for which classical physics offers no explanation. Since charges exist that charge is la carga las cargas existen en paquetes discretos, se dice packets, we say cuantizada; quantizedthe fundamen exist charge quantization in discrete que la carga est fundamental e se llama el cuantum called the As, cualquier ecantidad deany amountse charge that is ever de carga. quantum of charge. charge quantization in discrete packets, we

Ohanian Fisica 3e Capitulo Muestra 22

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