EletromagnetismoNewton Mansur

Cargas em movimento

𝐶𝑜𝑟𝑟𝑒𝑛𝑡𝑒 𝑖 =Δ𝑞

Δ𝑡

𝐷𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 𝑐𝑜𝑟𝑟𝑒𝑛𝑡𝑒 𝑗 =𝑖

𝑆

𝑅𝑒𝑠𝑖𝑠𝑡ê𝑛𝑐𝑖𝑎 𝑜ℎ𝑚𝑖𝑐𝑎 𝑅 =Δ𝑉

𝑖

Δ𝑙

𝑆

𝑣𝑜𝑙 = 𝑆Δ𝑙

Δ𝑞 = 𝜌𝑣𝑆Δ𝑙

Δ𝑙

𝑢

𝑢 =Δ𝑙

Δ𝑡

𝑖 =Δ𝑞

Δ𝑡=𝜌𝑣𝑆Δ𝑙

Δ𝑡= 𝜌𝑣𝑆u = jS

𝑗 = 𝜌𝑣u

𝜌𝑣 = 𝑛𝑒

𝑗 = 𝑛𝑒u

𝐸𝑓𝑒𝑖𝑡𝑜 𝐽𝑜𝑢𝑙𝑒 𝑃 = 𝑉𝑖

𝑉𝑒𝑙𝑜𝑐𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 𝑎𝑟𝑟𝑎𝑠𝑡𝑜 𝑑𝑒𝑟𝑖𝑣𝑎 𝑢

𝐷𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 𝐶𝑎𝑟𝑔𝑎 𝜌𝑣

𝐷𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 Á𝑡𝑜𝑚𝑜𝑠 𝑒𝑙é𝑡𝑟𝑜𝑛𝑠 𝑛

𝐶𝑜𝑟𝑟𝑒𝑛𝑡𝑒 𝑖 =Δ𝑞

Δ𝑡

𝐷𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 𝑐𝑜𝑟𝑟𝑒𝑛𝑡𝑒 𝑗 =𝑖

𝑆

𝑅𝑒𝑠𝑖𝑠𝑡ê𝑛𝑐𝑖𝑎 ôℎ𝑚𝑖𝑐𝑎 𝑅 =Δ𝑉

𝑖

𝐸𝑓𝑒𝑖𝑡𝑜 𝐽𝑜𝑢𝑙𝑒 𝑃 = 𝑉𝑖

𝑉𝑒𝑙𝑜𝑐𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 𝑎𝑟𝑟𝑎𝑠𝑡𝑜 𝑑𝑒𝑟𝑖𝑣𝑎 𝑢

𝐷𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 𝐶𝑎𝑟𝑔𝑎 𝜌𝑣

𝐷𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 Á𝑡𝑜𝑚𝑜𝑠 𝑒𝑙é𝑡𝑟𝑜𝑛𝑠 𝑛

𝑗 = 𝑛𝑒u

𝑚𝑢

τ= 𝐹 = 𝑒𝐸

𝑢 =𝑒τ

𝑚𝐸 =

𝑗

𝑛𝑒

𝑗 =𝑛𝑒2τ

𝑚𝐸

𝑗 = 𝜎𝐸

𝜎 =𝑛𝑒2τ

𝑚

𝐶𝑜𝑛𝑑𝑢𝑡𝑖𝑣𝑖𝑑𝑎𝑑𝑒 𝑒𝑙é𝑡𝑟𝑖𝑐𝑎 𝜎

𝑖 =Δ𝑉

𝑅

𝑖

𝑆= 𝑗 =

Δ𝑉

𝑆𝑅

𝑖

𝑆= 𝑗 =

𝑙

𝑆𝑅

Δ𝑉

𝑙=

𝑙

𝑆𝑅𝐸

𝑗 =𝑙

𝑆𝑅𝐸 = 𝜎𝐸

𝜌 =𝑆𝑅

𝑙

𝜌 =1

𝜎

𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑣𝑖𝑑𝑎𝑑𝑒 𝑒𝑙é𝑡𝑟𝑖𝑐𝑎 𝜌

𝐶𝑜𝑟𝑟𝑒𝑛𝑡𝑒 𝑖 =Δ𝑞

Δ𝑡

𝐷𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 𝑐𝑜𝑟𝑟𝑒𝑛𝑡𝑒 𝑗 =𝑖

𝑆

𝑅𝑒𝑠𝑖𝑠𝑡ê𝑛𝑐𝑖𝑎 ôℎ𝑚𝑖𝑐𝑎 𝑅 =Δ𝑉

𝑖

𝐸𝑓𝑒𝑖𝑡𝑜 𝐽𝑜𝑢𝑙𝑒 𝑃 = 𝑉𝑖

𝑉𝑒𝑙𝑜𝑐𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 𝑎𝑟𝑟𝑎𝑠𝑡𝑜 𝑑𝑒𝑟𝑖𝑣𝑎 𝑢

𝐷𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 𝐶𝑎𝑟𝑔𝑎 𝜌𝑣

𝐷𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 Á𝑡𝑜𝑚𝑜𝑠 𝑒𝑙é𝑡𝑟𝑜𝑛𝑠 𝑛

𝑗 = 𝑛𝑒u

𝑗 = 𝜎𝐸

𝐶𝑜𝑛𝑑𝑢𝑡𝑖𝑣𝑖𝑑𝑎𝑑𝑒 𝑒𝑙é𝑡𝑟𝑖𝑐𝑎 𝜎

𝜌 =1

𝜎

𝑅𝑒𝑠𝑖𝑠𝑡𝑖𝑣𝑖𝑑𝑎𝑑𝑒 𝑒𝑙é𝑡𝑟𝑖𝑐𝑎 𝜌

𝐶𝑜𝑏𝑟𝑒

𝑑𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 𝑚𝑎𝑠𝑠𝑎 8920 𝑘𝑔/𝑚3

𝑀𝑎𝑠𝑠𝑎 𝑎𝑡ô𝑚𝑖𝑐𝑎 64

𝑁ú𝑚𝑒𝑟𝑜 𝑑𝑒 𝐴𝑣𝑜𝑔𝑎𝑑𝑟𝑜 6,022𝑥1023

64 𝑘𝑔 𝑑𝑒 𝑐𝑜𝑏𝑟𝑒 − 6,022𝑥1026 á𝑡𝑜𝑚𝑜𝑠

𝑑𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 𝑚𝑜𝑙𝑒𝑠 =8920

64= 139 𝑚𝑜𝑙𝑒𝑠/

𝑑𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 á𝑡𝑜𝑚𝑜𝑠 = 139𝑥6,022𝑥1026

𝑛 = 8,37𝑥1029 á𝑡𝑜𝑚𝑜𝑠/𝑚3

𝑢 =𝑗

𝑛𝑒

𝑖 = 1𝐴 𝑆 = 1𝑚𝑚2 𝑗 = 1𝑥106𝐴/𝑚2

=1𝑥106

8,37𝑥1029𝑥1,602𝑥10−19= 7,5𝑥10−6𝑚/𝑠

𝑢 = 2,7𝑐𝑚/ℎ

Ԧ𝑣𝐵

Ԧ𝐹

Ԧ𝐹 ∝ 𝑞 Ԧ𝑣 𝐵

Ԧ𝐹 ⊥ 𝐵

Ԧ𝐹 ⊥ Ԧ𝑣

Ԧ𝐹 = 𝑚 Ԧ𝑔

Ԧ𝐹 = 𝑞𝐸

Ԧ𝐹 = 𝑞 Ԧ𝑣 × 𝐵

Ԧ𝑣𝐵

Ԧ𝑣 × 𝐵

Ԧ𝑣

Ԧ𝐹

Ԧ𝐹+ Ԧ𝐹−

Ԧ𝐹 = 𝑞 Ԧ𝑣 × 𝐵

𝐹𝑀 = 𝑞𝑣𝐵 = 𝑚𝑣2

𝑅

𝑅 =𝑚𝑣

𝑞𝐵𝑅

𝐵

𝐼

𝒅𝒒

𝒗

𝑑 Ԧ𝐹 = 𝑑𝑞 Ԧ𝑣 × 𝐵

𝑑 Ԧ𝐹

𝒅Ԧ𝒍

Ԧ𝑣 =𝑑Ԧ𝑙

𝑑𝑡𝑑 Ԧ𝐹 = 𝑑𝑞

𝑑Ԧ𝑙

𝑑𝑡× 𝐵 𝑑 Ԧ𝐹 =

𝑑𝑞

𝑑𝑡𝑑Ԧ𝑙 × 𝐵

𝑑 Ԧ𝐹 = 𝐼𝑑Ԧ𝑙 × 𝐵 𝑑𝑞 Ԧ𝑣 ≡ 𝐼𝑑Ԧ𝑙

𝐵𝐼

𝑑 Ԧ𝐹

𝒅Ԧ𝒍

𝑑 Ԧ𝐹 = 𝐼𝑑Ԧ𝑙 × 𝐵

𝑑 Ԧ𝐹

𝒅Ԧ𝒍𝑑 Ԧ𝐹

𝒅Ԧ𝒍𝑑 Ԧ𝐹

𝒅Ԧ𝒍𝐿

Ԧ𝐹 = 𝐼𝐿 × 𝐵

𝐵

𝐼

𝑑 Ԧ𝐹

𝒅Ԧ𝒍

𝑑 Ԧ𝐹 = 𝐼𝑑Ԧ𝑙 × 𝐵

𝑑 Ԧ𝐹𝐻

𝑑 Ԧ𝐹𝑉 𝑑Ԧ𝐹

𝒅Ԧ𝒍𝑑 Ԧ𝐹𝐻

𝑑 Ԧ𝐹𝑉

𝑑𝐹 = 𝐼𝑑𝑙𝐵 𝑑𝐹𝐻 = 𝐼𝑑𝑙𝐵𝑐𝑜𝑠𝜃

𝜃

𝜃

𝑑𝐹𝑉 = 𝐼𝑑𝑙𝐵𝑠𝑒𝑛𝜃

𝑑𝜃

𝑑𝑙 = 𝑅𝑑𝜃

𝑑𝐹𝑉 = 𝐼𝑅𝐵𝑠𝑒𝑛𝜃𝑑𝜃

𝐹𝑉 = 𝐼𝑅𝐵න0

𝜋

𝑠𝑒𝑛𝜃𝑑𝜃 𝐹𝑉 = 2𝐼𝑅𝐵

𝑅

𝐵

𝐼

𝑑 Ԧ𝐹

𝒅Ԧ𝒍

𝑑 Ԧ𝐹 = 𝐼𝑑Ԧ𝑙 × 𝐵

𝑑 Ԧ𝐹𝐻

𝑑 Ԧ𝐹𝑉

𝑑𝐹 = 𝐼𝑑𝑙𝐵 𝑑𝐹𝐻 = 𝐼𝑑𝑙𝐵𝑐𝑜𝑠𝜃

𝜃

𝜃

𝑑𝐹𝑉 = 𝐼𝑑𝑙𝐵𝑠𝑒𝑛𝜃

𝑑𝜃

𝑑𝑙 = 𝑅𝑑𝜃

𝑑𝐹𝑉 = 𝐼𝑅𝐵𝑠𝑒𝑛𝜃𝑑𝜃

𝐹𝑉 = 𝐼𝑅𝐵න0

𝛼

𝑠𝑒𝑛𝜃𝑑𝜃 𝐹𝑉 = 𝐼𝑅𝐵(1 − 𝑐𝑜𝑠𝛼)

𝛼

𝐹𝐻 = 𝐼𝑅𝐵𝑠𝑒𝑛𝛼

𝐼

𝐼

𝐼

𝐼

𝐵

Ԧ𝐹1

− Ԧ𝐹1

Ԧ𝐹2− Ԧ𝐹2

𝑎

𝑏

Ԧ𝐹 = 𝐼𝐿 × 𝐵 𝐹1 = 𝐼𝑎𝐵

𝐹2 = 𝐼𝑏𝐵

𝐹𝑇 = 0

𝐼

𝐼𝐵

Ԧ𝐹1

− Ԧ𝐹1

Ԧ𝐹2

Ԧ𝐹 = 𝐼𝐿 × 𝐵 𝐹1 = 𝐼𝑎𝐵

𝐹2 = 𝐼𝑏𝐵𝑠𝑒𝑛𝜃

𝐹𝑇 = 0

𝜃

Ԧ𝑟

Ԧ𝜏 = Ԧ𝑟 × Ԧ𝐹

𝛼

𝜏 = 𝑟𝐹𝑠𝑒𝑛𝛼 =𝑏

2𝐹1𝑠𝑒𝑛𝛼 =

1

2𝐼𝑎𝑏𝐵𝑠𝑒𝑛𝛼

𝜏𝑇 = 𝐼𝑎𝑏𝐵𝑠𝑒𝑛𝛼 = 𝐼𝑆𝐵𝑠𝑒𝑛𝛼

Ԧ𝑆

𝛼

Ԧ𝜏𝑇 = 𝐼 Ԧ𝑆 × 𝐵 Ԧ𝜇 = 𝐼 Ԧ𝑆 Ԧ𝜏𝑇 = Ԧ𝜇 × 𝐵

Ԧ𝜇 − 𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑑𝑖𝑝𝑜𝑙𝑜 𝑚𝑎𝑔𝑛é𝑡𝑖𝑐𝑜

Ԧ𝜇

𝑃𝑎𝑟𝑎 𝑜 𝑑𝑖𝑝𝑜𝑙𝑜 𝑒𝑙é𝑡𝑟𝑖𝑐𝑜 ර𝐸𝑑 Ԧ𝑆 = 0 𝛻. 𝐸 = 0

𝑃𝑎𝑟𝑎 𝑜 𝑑𝑖𝑝𝑜𝑙𝑜 𝑚𝑎𝑔𝑛é𝑡𝑖𝑐𝑜 ර𝐵𝑑 Ԧ𝑆 = 0 𝛻. 𝐵 = 0

𝐼

𝐼

𝐼

𝐼

𝐵

𝐼

𝐼𝐵

Ԧ𝐹1

− Ԧ𝐹1

Ԧ𝐹2

𝜃

Ԧ𝑟

𝛼

𝐵

𝛼

Ԧ𝜏𝑇 = Ԧ𝜇 × 𝐵

𝐵

𝐵

𝐵

𝑵

𝑺

𝑵

𝑺

𝑵

𝑺

𝑺

𝑵

𝑵

𝑺

𝑵

𝑺

𝑭

𝑭

𝐼

𝑰

𝐼

𝑵

𝑺

𝑵

𝑵

𝑵

𝑺

𝑺

𝑺

𝐵

𝐼

𝐵

𝐵

𝐵

𝐵

𝐵

𝑰

𝑑𝐵𝐼

𝒅𝒒

𝒗

𝑑𝐵𝛼 𝑑𝑞 Ԧ𝑣 × Ԧ𝑟

𝒅Ԧ𝒍

Ԧ𝑟

𝑰

X

𝐵 ⊥ Ԧ𝑣

𝐵 ⊥ Ԧ𝑟

Ƹ𝑟

𝑑𝐵𝛼 𝑑𝑞 Ԧ𝑣 × Ƹ𝑟

𝑑𝐵𝛼𝑑𝑞 Ԧ𝑣 × Ƹ𝑟

𝑟2

𝑑𝐵 =𝜇04𝜋

𝑑𝑞 Ԧ𝑣 × Ƹ𝑟

𝑟2

𝑑𝑞 Ԧ𝑣 ≡ 𝐼𝑑Ԧ𝑙

𝑑𝐵 =𝜇0𝐼

4𝜋

𝑑Ԧ𝑙 × Ƹ𝑟

𝑟2𝑑𝐵 =

𝜇0𝐼

4𝜋

𝑑Ԧ𝑙 × Ԧ𝑟

𝑟3

𝐿𝑒𝑖 𝑑𝑒 𝐵𝑖𝑜𝑡 − 𝑆𝑎𝑣𝑎𝑟𝑡

𝐵

𝐼𝒅Ԧ𝒍

𝛼

𝑑𝐵 =𝜇0𝐼

4𝜋

𝑑Ԧ𝑙 × Ƹ𝑟

𝑟2𝑑Ԧ𝑙 ⊥ Ԧ𝑟

Ԧ𝑟𝑅

𝑑𝐵 =𝜇0𝐼

4𝜋

𝑑𝑙

𝑅2𝐵 =

𝜇0𝐼

4𝜋

1

𝑅2න𝑑𝑙

𝐵 =𝜇0𝐼

4𝜋

𝑅𝛼

𝑅2𝐵 =

𝜇0𝐼

4𝜋𝑅𝛼

𝐼𝒅𝒙

Ԧ𝑟

𝑑𝐵 =𝜇0𝐼

4𝜋

𝑑 Ԧ𝑥 × Ƹ𝑟

𝑟2

𝒚

𝒙𝑥

𝑦

𝑑𝐵

𝜃

𝑑𝐵 =𝜇0𝐼

4𝜋

𝑑𝑥 Ƹ𝑟 𝑠𝑒𝑛𝜃

𝑟2𝑑𝐵 =

𝜇0𝐼

4𝜋

𝑑𝑥

𝑟2𝑦

𝑟𝑑𝐵 =

𝜇0𝐼𝑦

4𝜋

𝑑𝑥

𝑥2 + 𝑦2 ൗ32

𝐵 =𝜇0𝐼𝑦

4𝜋න−𝑎

𝑏 𝑑𝑥

𝑥2 + 𝑦2 ൗ32

0

𝑎 𝑏

𝐵 =𝜇0𝐼

2𝜋𝑦

𝐵 =𝜇0𝐼𝑦

4𝜋

𝑥

𝑦2 𝑥2 + 𝑦2𝑏−𝑎

𝑃𝑎𝑟𝑎 𝑜 𝑓𝑖𝑜 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑜𝑎 → ∞

𝑏 → ∞

𝐼

𝑟

𝐵

𝐵 =𝜇0𝐼

2𝜋𝑟

𝐼

𝑟

𝐵

𝐵 =𝜇0𝐼

2𝜋𝑟𝐵2𝜋𝑟 = 𝜇0𝐼

𝐼

𝐵1 𝐵2𝜋𝑟 = 𝜇0𝐼

𝑠1

𝑠2𝐵2

𝐵1𝑠1 = 𝐵2𝑠2 =𝜇0𝐼

𝑁

𝐵1𝑠1 + 𝐵2𝑠2 +⋯+ 𝐵𝑁𝑠𝑁 = 𝜇0𝐼

𝑑 Ԧ𝑠

න𝑠1

𝐵. 𝑑Ԧ𝑠 + න𝑠2

𝐵. 𝑑 Ԧ𝑠 + ⋯+න𝑠𝑁

𝐵. 𝑑 Ԧ𝑠 = 𝜇0𝐼

𝑑Ԧ𝑠

𝑑 Ԧ𝑠

𝑑Ԧ𝑠

𝐵

𝑁𝑜𝑠 𝑎𝑟𝑐𝑜𝑠

𝑁𝑎 𝑝𝑎𝑟𝑡𝑒 𝑟𝑎𝑑𝑖𝑎𝑙 න 𝐵. 𝑑 Ԧ𝑠 = 0𝐵 ⊥ 𝑑Ԧ𝑠

𝐵 ∥ 𝑑Ԧ𝑠

ර𝐵. 𝑑 Ԧ𝑠 = 𝜇0𝐼

𝑑Ԧ𝑠

𝐵. 𝑑 Ԧ𝑠

𝐼

ර𝐵. 𝑑 Ԧ𝑠 = 𝜇0𝐼

𝐵

𝐵

𝐼

න𝐴

𝐵. 𝑑 Ԧ𝑠 + න𝐵

𝐵. 𝑑 Ԧ𝑠 = 𝜇0𝐼

𝑑 Ԧ𝑠

ර𝐵. 𝑑 Ԧ𝑠 = 0

𝐴

𝐵

𝐶

න𝐴

𝐵. 𝑑 Ԧ𝑠 + න𝐶

𝐵. 𝑑 Ԧ𝑠 = 𝜇0𝐼

න𝐵

𝐵. 𝑑 Ԧ𝑠 = න𝐶

𝐵. 𝑑 Ԧ𝑠

𝐵

𝐵

𝑑 Ԧ𝑠𝑑 Ԧ𝑠න𝐶

𝐵. 𝑑 Ԧ𝑠 > 0

න𝐵

𝐵. 𝑑 Ԧ𝑠 < 0

න𝐵

𝐵. 𝑑 Ԧ𝑠 + න𝐶

𝐵. 𝑑 Ԧ𝑠 = 0

𝐼

ර𝐵. 𝑑Ԧ𝑠 = 𝜇0𝐼𝐼𝑛𝑡

𝐵

𝐵

𝐼𝐼𝑛𝑡

𝐼𝐸𝑥𝑡

𝐿𝑒𝑖 𝑑𝑒 𝐴𝑚𝑝è𝑟𝑒

ර𝐵. 𝑑Ԧ𝑙 = 𝜇0𝐼𝐼𝑛𝑡

𝐼

𝑟

ර𝐸. 𝑑Ԧ𝑙 = 0

𝐼

𝑟

𝐵

𝐼

𝑟

𝐵

𝐵2𝜋𝑟 = 𝜇0𝐼𝐼𝑛𝑡

ර𝐵. 𝑑Ԧ𝑙 = 𝜇0𝐼𝐼𝑛𝑡

𝑑Ԧ𝑙

𝐵 ∥ 𝑑Ԧ𝑙

𝐵 =𝜇0𝐼𝐼𝑛𝑡2𝜋𝑟

𝐵 =𝜇0𝐼

2𝜋𝑟

𝐼

𝑟

𝐵

𝐼

𝑟𝐵

𝐵2𝜋𝑟 = 𝜇0𝐼𝐼𝑛𝑡

ර𝐵. 𝑑Ԧ𝑙 = 𝜇0𝐼𝐼𝑛𝑡

𝑑Ԧ𝑙

𝐵 ∥ 𝑑Ԧ𝑙

𝐵 =𝜇0𝐼𝐼𝑛𝑡2𝜋𝑟

𝐵 =𝜇0𝐼

2𝜋𝑟

𝑑𝐵

𝑑𝐵

𝐼𝑟

𝐵

𝐼

𝑟𝐵

𝐵2𝜋𝑟 = 𝜇0𝐼𝐼𝑛𝑡

ර𝐵. 𝑑Ԧ𝑙 = 𝜇0𝐼𝐼𝑛𝑡

𝑑Ԧ𝑙

𝐵 ∥ 𝑑Ԧ𝑙

𝐵 =𝜇0𝐼𝐼𝑛𝑡2𝜋𝑟

𝑑𝐵

𝑑𝐵

𝐼𝐼𝑛𝑡𝐼

=𝐴𝐼𝑛𝑡𝐴

𝐼𝐼𝑛𝑡 = 𝑗𝐴𝐼𝑛𝑡

𝐼𝐼𝑛𝑡 = 𝐼𝜋𝑟2

𝜋𝑅2

𝐵 =𝜇0𝐼

2𝜋

𝑟

𝑅2

𝐷 = 𝜀𝐸 𝐷 =𝑞

4𝜋𝑟2ො𝑎𝑟

𝐻 =𝑞 Ԧ𝑣 × Ƹ𝑟

4𝜋𝑟2𝐻 =𝐵

𝜇𝑑𝐻 =

𝐼𝑑Ԧ𝑙 × Ƹ𝑟

4𝜋𝑟2

ර𝐻. 𝑑Ԧ𝑙 = 𝐼𝐼𝑛𝑡

𝐻 − 𝐶𝑎𝑚𝑝𝑜 𝑀𝑎𝑔𝑛é𝑡𝑖𝑐𝑜

𝐵 − 𝐷𝑒𝑛𝑠𝑖𝑑𝑎𝑑𝑒 𝑑𝑒 𝐹𝑙𝑢𝑥𝑜 𝑜𝑢 𝐼𝑛𝑑𝑢çã𝑜 𝑀𝑎𝑔𝑛é𝑡𝑖𝑐𝑎

𝜇 − 𝑃𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑑𝑎𝑑𝑒 𝑀𝑎𝑔𝑛é𝑡𝑖𝑐𝑎

𝑥

𝑦

𝑧

1 2

34

ර𝐻. 𝑑Ԧ𝑙

𝐻0𝑦

𝐻0𝑥

𝐻0

𝐻0 = 𝐻0𝑥 Ԧ𝑎𝑥 + 𝐻0𝑦 Ԧ𝑎𝑦 + 𝐻0𝑧 Ԧ𝑎𝑧

𝐻0𝑧

1 → 2 𝐻. ∆Ԧ𝑙 = 𝐻𝑦1→2∆𝑦

𝐻𝑦

= (𝐻0𝑦 +𝜕𝐻𝑦

𝜕𝑥

∆𝑥

2)∆𝑦

3 → 4 𝐻. ∆Ԧ𝑙 = 𝐻𝑦3→4(−∆𝑦) = (𝐻0𝑦 −𝜕𝐻𝑦

𝜕𝑥

∆𝑥

2)(−∆𝑦)

4 → 1 𝐻. ∆Ԧ𝑙 = 𝐻𝑥4→1∆x = (𝐻0𝑥 −𝜕𝐻𝑥𝜕𝑦

∆𝑦

2)∆𝑥

2 → 3 𝐻. ∆Ԧ𝑙 = 𝐻𝑦2→3(−∆𝑥) = (𝐻0𝑥 +𝜕𝐻𝑥𝜕𝑦

∆𝑦

2)(−∆𝑥)

1 → 2 → 3 → 4 𝐻. ∆Ԧ𝑙 =𝜕𝐻𝑦

𝜕𝑥−𝜕𝐻𝑥𝜕𝑦

∆𝑥∆𝑦

𝑥

𝑦

𝑧

1 2

34

ර𝐻. 𝑑Ԧ𝑙

𝐻0𝑦

𝐻0𝑥

𝐻0𝐻0𝑧

𝐻𝑦

𝐻.∆Ԧ𝑙 =𝜕𝐻𝑦

𝜕𝑥−𝜕𝐻𝑥𝜕𝑦

𝑆𝑧

∆𝑥 → 0 ∆𝑦 → 0

𝐻.𝑑Ԧ𝑙 =𝜕𝐻𝑦

𝜕𝑥−𝜕𝐻𝑥𝜕𝑦

𝑑𝑆𝑧

ර𝐻. 𝑑Ԧ𝑙 = න𝜕𝐻𝑦

𝜕𝑥−𝜕𝐻𝑥𝜕𝑦

𝑑𝑆𝑧

ර𝐻. 𝑑Ԧ𝑙 = න𝜕𝐻𝑦

𝜕𝑥−𝜕𝐻𝑥𝜕𝑦

Ԧ𝑎𝑧. 𝑑𝑆𝑧 Ԧ𝑎𝑧

ර𝐻. 𝑑Ԧ𝑙 = න𝜕𝐻𝑧𝜕𝑦

−𝜕𝐻𝑦

𝜕𝑧Ԧ𝑎𝑥 . 𝑑𝑆𝑥 Ԧ𝑎𝑥

ර𝐻. 𝑑Ԧ𝑙 = න𝜕𝐻𝑥𝜕𝑧

−𝜕𝐻𝑧𝜕𝑥

Ԧ𝑎𝑦 . 𝑑𝑆𝑦 Ԧ𝑎𝑦

𝑥

𝑦

𝑧

𝑑 Ԧ𝑆

𝐻

ර𝐻. 𝑑Ԧ𝑙 = න𝜕𝐻𝑧𝜕𝑦

−𝜕𝐻𝑦

𝜕𝑧Ԧ𝑎𝑥 +

𝜕𝐻𝑥𝜕𝑧

−𝜕𝐻𝑧𝜕𝑥

Ԧ𝑎𝑦 +𝜕𝐻𝑦

𝜕𝑥−𝜕𝐻𝑥𝜕𝑦

Ԧ𝑎𝑧 . 𝑑 Ԧ𝑆

𝑉

𝑉 =𝜕𝐻𝑧𝜕𝑦

−𝜕𝐻𝑦

𝜕𝑧Ԧ𝑎𝑥 +

𝜕𝐻𝑥𝜕𝑧

−𝜕𝐻𝑧𝜕𝑥

Ԧ𝑎𝑦 +𝜕𝐻𝑦

𝜕𝑥−𝜕𝐻𝑥𝜕𝑦

Ԧ𝑎𝑧

𝑉 = 𝛻 × 𝐻

ර𝐻. 𝑑Ԧ𝑙 = න 𝛻 × 𝐻 . 𝑑 Ԧ𝑆

ර𝐻. 𝑑Ԧ𝑙 = න𝑉. 𝑑 Ԧ𝑆

ර𝐻. 𝑑Ԧ𝑙 = 𝐼𝐼𝑛𝑡 න 𝛻 × 𝐻 . 𝑑Ԧ𝑆 = 𝐼𝐼𝑛𝑡 = න Ԧ𝑗. 𝑑 Ԧ𝑆

Ԧ𝑗

𝛻 × 𝐻 = Ԧ𝑗

ර𝐵. 𝑑Ԧ𝑙 = 𝜇0𝐼𝐼𝑛𝑡

ර𝐸. 𝑑Ԧ𝑙 = 0

ර𝐸. 𝑑 Ԧ𝑆 =𝑞

𝜀0

ර𝐵. 𝑑 Ԧ𝑆 = 0

𝛻. 𝐸 =𝜌

𝜀0

𝛻. 𝐵 = 0

𝛻 × 𝐵 = 𝜇0Ԧ𝑗

𝛻 × 𝐸 = 0

𝛻.𝐷 = 𝜌

𝛻.𝐻 = 0

𝛻 × 𝐻 = Ԧ𝑗

𝛻 × 𝐷 = 0

𝛻 × 𝐸 = 0 𝛻 × 𝐷 = 0

𝛻. 𝐵 = 0

𝛻 × 𝐸 = 0 𝛻 × 𝛻𝜑 = 0 𝐸 = −𝛻𝜑

𝜑 Ԧ𝑟 = 𝜑𝑃 Ԧ𝑟 + 𝜑0 𝛻𝜑 Ԧ𝑟 = 𝛻𝜑𝑃 Ԧ𝑟

𝛻. (𝛻 × Ԧ𝐴) = 0 𝐵 = 𝛻 × Ԧ𝐴

𝛻 × 𝛻𝛿 = 0 Ԧ𝐴 Ԧ𝑟 = Ԧ𝐴𝑃 Ԧ𝑟 + 𝛻𝛿 𝛻 × Ԧ𝐴 Ԧ𝑟 = 𝛻 × Ԧ𝐴𝑃 Ԧ𝑟

Ԧ𝐴 Ԧ𝑟 − 𝑃𝑜𝑡𝑒𝑛𝑐𝑖𝑎𝑙 𝑉𝑒𝑡𝑜𝑟

𝛻. 𝐸 =𝜌

𝜀0𝐸 = −𝛻𝜑 𝛻. (−𝛻𝜑) =

𝜌

𝜀0𝛻2𝜑 = −

𝜌

𝜀0

𝛻 × 𝐵 = 𝜇0Ԧ𝑗 𝐵 = 𝛻 × Ԧ𝐴

𝛻 × 𝛻 × Ԧ𝐴 = 𝛻. 𝛻. Ԧ𝐴 − 𝛻2 Ԧ𝐴 Ԧ𝐴 Ԧ𝑟 = Ԧ𝐴𝑃 Ԧ𝑟 + 𝛻𝛿

𝛻. Ԧ𝐴 Ԧ𝑟 = 𝛻. Ԧ𝐴𝑃 Ԧ𝑟 + 𝛻. 𝛻𝛿 𝛻. Ԧ𝐴 Ԧ𝑟 = 𝛻. Ԧ𝐴𝑃 Ԧ𝑟 + 𝛻2𝛿

𝛻. Ԧ𝐴𝑃 Ԧ𝑟 = −𝛻2𝛿 𝛻. Ԧ𝐴 Ԧ𝑟 = 0

𝛻 × 𝛻 × Ԧ𝐴 = −𝛻2 Ԧ𝐴 = 𝛻 × 𝐵 = 𝜇0Ԧ𝑗 𝛻2 Ԧ𝐴 = −𝜇0Ԧ𝑗

Teorema de Helmholtz

1

4𝜋𝛻2න

Ԧ𝐺(Ԧ𝑟′)

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ =

1

4𝜋න Ԧ𝐺(Ԧ𝑟′)𝛻2

1

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ = න Ԧ𝐺(Ԧ𝑟′)𝛿(Ԧ𝑟 − Ԧ𝑟′)𝑑𝑣′ = Ԧ𝐺(Ԧ𝑟)

𝑈 𝑟 =1

4𝜋න

Ԧ𝐺(Ԧ𝑟′)

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ 𝛻2𝑈 𝑟 =

1

4𝜋𝛻2න

Ԧ𝐺(Ԧ𝑟′)

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′

𝛻2𝑈 𝑟 = Ԧ𝐺 𝑟 𝑈 Ԧ𝑟 =1

4𝜋න𝛻′2𝑈 Ԧ𝑟′

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ 𝑊 Ԧ𝑟 =

1

4𝜋න𝛻′2𝑊 Ԧ𝑟′

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′

Ԧ𝐹 = −𝛻𝑊 + 𝛻 × 𝑈 𝛻 ∙ Ԧ𝐹 = −𝛻2𝑊

𝛻 × Ԧ𝐹 = 𝛻 × 𝛻 × 𝑈 = 𝛻 ∙ 𝛻 ∙ 𝑈 − 𝛻2𝑈 𝛻 ∙ 𝑈 = 0 𝛻 × Ԧ𝐹 = −𝛻2𝑈

𝛻 ∙ Ԧ𝐹 = −𝛻2𝑊 = 𝐷

𝛻 × Ԧ𝐹 = −𝛻2𝑈 = Ԧ𝐺

𝐸 Ԧ𝑟 =1

4𝜋න𝛻′2𝐸 Ԧ𝑟′

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ 𝐸 = −𝛻𝑉 + 𝛻 × 𝑈

𝐵 Ԧ𝑟 =1

4𝜋න𝛻′2𝐵 Ԧ𝑟′

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ 𝐵 = −𝛻𝑊 + 𝛻 × Ԧ𝐴

𝐸 =1

4𝜋න𝛻′2𝐸 Ԧ𝑟′

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′

𝐸 = −𝛻. 𝑉 𝑉 = −1

4𝜋න

𝛻. 𝐸

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ + 𝑉0 = −

1

4𝜋𝜀0න

𝜌

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ + 𝑉0

Ԧ𝐴 = −1

4𝜋න

𝛻 × 𝐵

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ + Ԧ𝐴0 = −

𝜇04𝜋

නԦ𝑗

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ + Ԧ𝐴0

𝐵 =1

4𝜋න𝛻′2𝐵 Ԧ𝑟′

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ =

1

4𝜋න𝛻. 𝛻. 𝐵 − 𝛻 × 𝛻 × 𝐵

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′

𝐵 = 𝛻 × Ԧ𝐴

𝑉 = −1

4𝜋𝜀0න𝜌

𝑟𝑑𝑣 + 𝑉0 Ԧ𝐴 = −

𝜇04𝜋

නԦ𝑗

𝑟𝑑𝑣 + Ԧ𝐴0

𝐻

1

2

𝐻1

∆𝑙

∆𝑙

∆ℎ∆ℎ

𝐻1𝑛

𝐻1𝑡

𝐻2𝐻2𝑛

𝐻2𝑡

𝐴

𝐵𝐶

𝐷

න𝐴−𝐵

𝐻. 𝑑Ԧ𝑙 = −𝐻2𝑛∆ℎ

2− 𝐻1𝑛

∆ℎ

2

න𝐵−𝐶

𝐻. 𝑑Ԧ𝑙 = − −𝐻2𝑡∆𝑙

න𝐶−𝐷

𝐻. 𝑑Ԧ𝑙 = − −𝐻1𝑛∆ℎ

2− −𝐻2𝑛

∆ℎ

2

න𝐷−𝐴

𝐻. 𝑑Ԧ𝑙 = −𝐻1𝑡∆𝑙

−𝐻2𝑛∆ℎ

2− 𝐻1𝑛

∆ℎ

2+ 𝐻2𝑡∆𝑙 + 𝐻2𝑛

∆ℎ

2+ 𝐻1𝑛

∆ℎ

2− 𝐻1𝑡∆𝑙 =

𝐼

𝐴

∆ℎ → 0 𝐻2𝑡 − 𝐻1𝑡 = 𝐾

ර𝐻. 𝑑Ԧ𝑙 = Ԧ𝑗

𝐻2𝑡 − 𝐻1𝑡 =𝐼

∆𝑙

𝐵2𝑡𝜇0

−𝐵1𝑡𝜇0

= 𝐾

𝐻

1

2

𝐵1𝐵1𝑛

𝐵1𝑡

𝐵2𝐵2𝑛

𝐵2𝑡

𝐵1𝑛

𝐵2𝑛

ර𝐵. 𝑑 Ԧ𝑆 = 0

𝐵2𝑛𝐴 − 𝐵1𝑛𝐴 = 0

𝐵2𝑛 − 𝐵1𝑛 = 0

𝐻2𝑛 − 𝐻1𝑛 = 0

𝐻2𝑡 − 𝐻1𝑡 = 𝐾

𝐵2𝑡 − 𝐵1𝑡 = 𝜇0𝐾

ℎ

ℎ → 0

Ԧ𝒋 Ԧ𝒋

Plano espesso infinito

d

Ԧ𝒋 Ԧ𝒋

Plano espesso infinito

d

𝒅𝑩

𝒅𝑩

𝑩

𝒅𝑩

𝒅𝑩

𝑩

𝑩

𝑩𝑩

𝑩

𝑩

𝒅Ԧ𝒍

𝒅Ԧ𝒍

𝒅Ԧ𝒍

𝒅Ԧ𝒍

𝒅Ԧ𝒍

𝒅Ԧ𝒍

d

L

ර𝐵. 𝑑Ԧ𝑙 = 𝜇0𝐼𝐼𝑛𝑡 2𝐵𝐿 = 𝜇0𝑗𝐿𝑑 𝐵 =1

2𝜇0𝑗𝑑

𝑩

Ԧ𝒋 Ԧ𝒋

Plano espesso infinito

d

𝒅𝑩

𝒅𝑩

𝑩

𝒅𝑩

𝒅𝑩

𝑩

𝑩𝑩

𝑩

𝒅Ԧ𝒍

𝒅Ԧ𝒍

𝒅Ԧ𝒍

𝒅Ԧ𝒍2y

L

y

x

ර𝐵. 𝑑Ԧ𝑙 = 𝜇0𝐼𝐼𝑛𝑡 2𝐵𝐿 = 𝜇0𝑗2𝑦𝐿 𝐵 = 𝜇0𝑗𝑦

𝐵 = −𝜇0𝑗𝑦 ො𝑥

Ԧ𝒋 Ԧ𝒋

Plano espesso infinito

d

y

𝛻2 Ԧ𝐴 = −𝜇0Ԧ𝑗Dentro

𝜕2𝐴𝑥𝜕𝑥2

+𝜕2𝐴𝑥𝜕𝑦2

+𝜕2𝐴𝑥𝜕𝑧2

ො𝑎𝑥 +𝜕2𝐴𝑦

𝜕𝑥2+𝜕2𝐴𝑦

𝜕𝑦2+𝜕2𝐴𝑦

𝜕𝑧2ො𝑎𝑦 +

𝜕2𝐴𝑧𝜕𝑥2

+𝜕2𝐴𝑧𝜕𝑦2

+𝜕2𝐴𝑧𝜕𝑧2

ො𝑎𝑧 = −𝜇0𝐽 ො𝑎𝑧

𝜕2𝐴𝑧𝜕𝑦2

= −𝜇0𝐽 𝐴𝑧 𝑦 = −𝜇0𝐽𝑦2

2+ 𝐶𝑦 + 𝐷 𝐵 = 𝛻 × Ԧ𝐴

𝛻 × Ԧ𝐴 =𝜕𝐴𝑧𝜕𝑦

ො𝑎𝑥 = −𝜇0𝐽𝑦 + 𝐶 ො𝑎𝑥Para y=0 B=0 => C=0

D=0 Referência

Ԧ𝐴 = −𝜇0𝐽𝑦2

2ො𝑎𝑧

𝑨

Ԧ𝒋 Ԧ𝒋

Plano espesso infinito

d

y

𝛻2 Ԧ𝐴 = 0Fora 𝜕2𝐴𝑧

𝜕𝑦2= 0 𝐴𝑧 𝑦 = 𝐸𝑦 + 𝐹

𝑨

𝐸𝑑

2+ 𝐹 = −𝜇0𝐽

𝑑2

8

No contorno

𝐴𝑧 𝐷𝑒𝑛𝑡𝑟𝑜 𝑦 →𝑑

2= 𝐴𝑧 𝐹𝑜𝑟𝑎 𝑦 →

𝑑

2

Ԧ𝐴 = −𝜇0𝐽𝑑

2𝑦 −

𝑑

4ො𝑎𝑧

𝑨

𝑨

𝜕𝐴𝑧 𝐷𝑒𝑛𝑡𝑟𝑜𝜕𝑦

𝑦 →𝑑

2=𝜕𝐴𝑧 𝐹𝑜𝑟𝑎

𝜕𝑦𝑦 →

𝑑

2𝐸 = −𝜇0𝐽

𝑑

2𝐹 =

1

8𝜇0𝐽𝑑

2

𝐼𝑟

𝐵

𝐼

𝑟

𝛻2 Ԧ𝐴 = −𝜇0Ԧ𝑗Dentro

1

𝜌

𝜕

𝜕𝜌𝜌𝜕𝐴𝑧𝜕𝑟

= −𝜇0𝑗 𝜌𝜕𝐴𝑧𝜕𝜌

= −𝜇0𝑗𝜌2

2+ 𝐶

𝐴𝑧 = −𝜇0𝑗𝜌2

4+ 𝐶𝑙𝑛𝜌 + 𝐷

𝑂 𝑝𝑜𝑡𝑒𝑛𝑐𝑖𝑎𝑙 𝑛ã𝑜 𝑑𝑖𝑣𝑒𝑟𝑔𝑒 𝑒𝑚 𝜌 = 0 𝑒𝑛𝑡ã𝑜 𝐶 = 0

𝑅𝑒𝑓𝑒𝑟ê𝑛𝑐𝑖𝑎 𝐷 = 0

𝐼𝑟

𝛻2 Ԧ𝐴 = 0Fora1

𝜌

𝜕

𝜕𝜌𝜌𝜕𝐴𝑧𝜕𝜌

= 0 𝜌𝜕𝐴𝑧𝜕𝜌

= 𝐸

𝐴𝑧 = 𝐸𝑙𝑛𝜌 + 𝐹

𝑁𝑜 𝑐𝑜𝑛𝑡𝑜𝑟𝑛𝑜 𝐴𝐷𝑒𝑛𝑡𝑟𝑜 𝑅 = 𝐴𝐹𝑜𝑟𝑎 𝑅 −𝜇0𝑗𝑅2

4= 𝐸𝑙𝑛𝑅 + 𝐹

𝑁𝑜 𝑐𝑜𝑛𝑡𝑜𝑟𝑛𝑜𝜕

𝜕𝜌𝐴𝐷𝑒𝑛𝑡𝑟𝑜 𝑅 =

𝜕

𝜕𝜌𝐴𝐹𝑜𝑟𝑎 𝑅 −𝜇0

𝑗𝑅

2=𝐸

𝑅

𝐸 = −𝜇0𝑗𝑅2

2𝐹 = 𝜇0

𝑗𝑅2

2(𝑙𝑛𝑅 −

1

2)

𝐼

𝑟

Fora Ԧ𝐴 = −𝜇0𝑗𝑅2

2𝑙𝑛

𝜌

𝑅+1

2Ƹ𝑧

Dentro Ԧ𝐴 = −𝜇0𝑗𝜌2

4Ƹ𝑧 𝐵 = 𝛻 × Ԧ𝐴 = −

𝜕𝐴𝑧𝜕𝜌

ො𝜑 = 𝜇0𝑗𝜌

2ො𝜑

𝐵 = 𝛻 × Ԧ𝐴 = −𝜕𝐴𝑧𝜕𝜌

ො𝜑 = 𝜇0𝑗𝑅2

2𝜌ො𝜑

xy

z

Ԧ𝑟

𝑥

𝑦

𝑧

xy

z

Ԧ𝑟′𝑥

𝑦

𝑧

Ԧ𝑟

𝑅

𝑉 =1

4𝜋𝜀0

𝑞

𝑟 𝑉 =1

4𝜋𝜀0න

𝜌(Ԧ𝑟′)

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ + 𝑉0𝛻2𝑉 = −

𝜌

𝜀0

𝛻2 Ԧ𝐴 = −𝜇0Ԧ𝑗 Ԧ𝐴 =𝜇04𝜋

නԦ𝐽( Ԧ𝑟′)

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′ + Ԧ𝐴0

Ԧ𝑗

Ԧ𝐴 =𝜇04𝜋

නԦ𝐽(Ԧ𝑟′)

Ԧ𝑟 − Ԧ𝑟′𝑑𝑣′

𝒅𝒛′

Ԧ𝑟

𝝆

𝒛𝑧’

𝜌

𝜃

0

𝑎

Ԧ𝐽 Ԧ𝑟′ =1

2𝜋𝜌′𝐼𝛿(𝜌′) Ƹ𝑧

න Ԧ𝐽 Ԧ𝑟′ 𝜌′𝑑𝜑′𝑑𝜌′ = 𝐼

Ԧ𝑟 − Ԧ𝑟′ = 𝜌2 + 𝑧′2 Ԧ𝐴 =𝜇04𝜋

නԦ𝐽(Ԧ𝑟′)

𝜌2 + 𝑧′2𝑑𝑣′Ԧ𝐽 Ԧ𝑟′ 𝑑𝑣′ =

1

2𝜋𝜌′𝐼𝛿(𝜌′)𝜌′𝑑𝜑′𝑑𝜌′dz′

Ԧ𝐴 =𝜇04𝜋

න−𝑎

𝑏 1

𝜌2 + 𝑧′2

1

2𝜋𝜌′𝐼𝛿(𝜌′)𝜌′𝑑𝜑′𝑑𝜌′dz′ Ƹ𝑧 =

𝜇04𝜋

𝐼 න−𝑎

𝑏 1

𝜌2 + 𝑧′2dz′ Ƹ𝑧

𝑏

Ԧ𝐴 =𝜇04𝜋

𝐼 𝑙𝑛 𝑧′ + 𝜌2 + 𝑧′2𝑏−𝑎

=𝜇04𝜋

𝐼 𝑙𝑛𝑏 + 𝜌2 + 𝑏2

−𝑎 + 𝜌2 + 𝑎2

𝑁𝑜𝑡𝑒 𝑞𝑢𝑒 𝑝𝑎𝑟𝑎 𝑢𝑚 𝑓𝑖𝑜 𝑠𝑒𝑚𝑖 − 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑜 𝑜𝑢 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑜 𝑛ã𝑜 𝑝𝑜𝑑𝑒𝑚𝑜𝑠 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑟𝑜 𝑝𝑜𝑡𝑒𝑛𝑐𝑖𝑎𝑙 𝑣𝑒𝑡𝑜𝑟 𝑑𝑒𝑠𝑡𝑎 𝑓𝑜𝑟𝑚𝑎

qz

R

x

y

z

𝑑𝐵 =𝜇0𝐼

4𝜋

𝑑Ԧ𝑙 × Ƹ𝑟

𝑟2

i

𝑃𝑜𝑟 𝑠𝑖𝑚𝑒𝑡𝑟𝑖𝑎 𝑠ó 𝑠𝑜𝑏𝑟𝑎 𝑑𝐵𝑧

Ԧ𝑟

𝑑Ԧ𝑙

𝑑𝐵𝑑𝐵𝑧

𝑪𝒂𝒎𝒑𝒐𝑴𝒂𝒈𝒏é𝒕𝒊𝒄𝒐 𝒅𝒆 𝒖𝒎𝒂 𝒆𝒔𝒑𝒊𝒓𝒂

𝑑𝐵 =𝜇0𝐼

4𝜋

𝑑𝑙

𝑟2𝑑Ԧ𝑙 ⊥ Ƹ𝑟 𝑑𝐵𝑧 =

𝜇0𝐼

4𝜋

𝑑𝑙

𝑟2𝑠𝑒𝑛𝜃

𝑑𝐵𝑧 =𝜇0𝐼

4𝜋

𝑑𝑙

𝑟2𝑅

𝑟=𝜇0𝐼

4𝜋

𝑅𝑑𝑙

𝑟3=𝜇0𝐼

4𝜋

𝑅𝑑𝑙

𝑅2 + 𝑧2 ൗ32

𝐵𝑧 = න𝜇0𝐼

4𝜋

𝑅𝑑𝑙

𝑅2 + 𝑧2 ൗ32

𝐵𝑧 =𝜇0𝐼

4𝜋

𝑅2𝜋𝑅

𝑅2 + 𝑧2 ൗ32=𝜇0𝐼

2

𝑅2

𝑅2 + 𝑧2 ൗ32

𝑑𝐵 =𝜇0𝐼

4𝜋

𝑑Ԧ𝑙 × Ԧ𝑟′

𝑟′3

𝑪𝒂𝒎𝒑𝒐𝑴𝒂𝒈𝒏é𝒕𝒊𝒄𝒐 𝒅𝒆 𝒖𝒎𝒂 𝒆𝒔𝒑𝒊𝒓𝒂

Ԧ𝑟′ = Ԧ𝑟 − 𝑅

Ԧ𝑟 = 𝑥 Ƹ𝑖 + 𝑦 Ƹ𝑗 + 𝑧𝑘

𝑅 = 𝑅𝑐𝑜𝑠𝜑 Ƹ𝑖 + 𝑅𝑠𝑒𝑛𝜑 Ƹ𝑗

x

y

z

i

Ԧ𝑟’

𝑑Ԧ𝑙

Ԧ𝑟

𝑅𝜑

Ԧ𝑟′ = (𝑥 − 𝑅𝑐𝑜𝑠𝜑) Ƹ𝑖 + (𝑦 − 𝑅𝑠𝑒𝑛𝜑) Ƹ𝑗 + 𝑧𝑘

𝑑Ԧ𝑙 = −𝑅𝑑𝜑𝑠𝑒𝑛𝜑 Ƹ𝑖 + 𝑅𝑑𝜑𝑐𝑜𝑠𝜑 Ƹ𝑗

𝑑Ԧ𝑙 × Ԧ𝑟′

= −𝑅𝑑𝜑𝑠𝑒𝑛𝜑 𝑦 − 𝑅𝑠𝑒𝑛𝜑 𝑘 − 𝑅𝑑𝜑𝑠𝑒𝑛𝜑𝑧 − Ƹ𝑗 + 𝑅𝑑𝜑𝑐𝑜𝑠𝜑 𝑥 − 𝑅𝑐𝑜𝑠𝜑 −𝑘

+ 𝑅𝑑𝜑𝑐𝑜𝑠𝜑 𝑧 Ƹ𝑖𝑑Ԧ𝑙 × Ԧ𝑟′ = 𝑅𝑧𝑐𝑜𝑠𝜑𝑑𝜑 Ƹ𝑖 + 𝑅𝑧𝑠𝑒𝑛𝜑𝑑𝜑 Ƹ𝑗 + 𝑅(𝑅 − 𝑦𝑠𝑒𝑛𝜑 − 𝑥𝑐𝑜𝑠𝜑)𝑑𝜑𝑘

𝑑𝐵 =𝜇0𝐼

4𝜋

𝑅𝑧𝑐𝑜𝑠𝜑𝑑𝜑 Ƹ𝑖 + 𝑅𝑧𝑠𝑒𝑛𝜑𝑑𝜑 Ƹ𝑗 + 𝑅(𝑅 − 𝑦𝑠𝑒𝑛𝜑 − 𝑥𝑐𝑜𝑠𝜑)𝑑𝜑𝑘

(𝑟2 + 𝑅2 − 2𝑥𝑅𝑐𝑜𝑠𝜑 − 2𝑦𝑅𝑠𝑒𝑛𝜑)32

𝑃𝑜𝑟 𝑐𝑎𝑢𝑠𝑎 𝑑𝑎 𝑠𝑖𝑚𝑒𝑡𝑟𝑖𝑎 𝑐𝑖𝑙í𝑛𝑑𝑟𝑖𝑐𝑎 𝑝𝑜𝑑𝑒𝑚𝑜𝑠 𝑔𝑖𝑟𝑎𝑟 𝑜 𝑒𝑖𝑥𝑜 𝑒𝑚 𝑡𝑜𝑟𝑛𝑜 𝑑𝑒 𝑧 𝑎𝑡é 𝑞𝑢𝑒𝑥 𝑜𝑢 𝑦 𝑠𝑒𝑗𝑎 𝑧𝑒𝑟𝑜, 𝑓𝑎𝑐𝑖𝑙𝑖𝑡𝑎𝑛𝑑𝑜 𝑎 𝑐𝑜𝑛𝑡𝑎

𝑪𝒂𝒎𝒑𝒐𝑴𝒂𝒈𝒏é𝒕𝒊𝒄𝒐 𝒅𝒆 𝒖𝒎𝒂 𝒆𝒔𝒑𝒊𝒓𝒂

Ԧ𝑟′ = Ԧ𝑟 − 𝑅

𝑟′2= 𝑟2 + 𝑅2 − 2𝑟𝑅𝑐𝑜𝑠𝜃

Ԧ𝐴 =𝜇04𝜋

නԦ𝑗

𝑟′𝑑𝑣′

Ԧ𝐴 =𝜇0𝑖

4𝜋ර

1

𝑟2 + 𝑅2 − 2𝑟𝑅𝑐𝑜𝑠𝜃𝑑Ԧ𝑙′q

x

y

z

i

Ԧ𝑟’

𝑑Ԧ𝑙′

Ԧ𝑟

𝑅𝜑

Ԧ𝑗𝑑𝑣′ = 𝑖𝑑Ԧ𝑙′

1

𝑟2 + 𝑅2 − 2𝑟𝑅𝑐𝑜𝑠𝜃=

1

𝑟𝑅𝑟

2

+ 1 − 2𝑅𝑟𝑐𝑜𝑠𝜃

=1

𝑟

𝑛=0

∞𝑅

𝑟

𝑛

𝑃𝑛(𝑐𝑜𝑠𝜃)

Ԧ𝐴 =𝜇0𝑖

4𝜋

𝑛=0

∞1

𝑟𝑛+1ර𝑅𝑛𝑃𝑛(𝑐𝑜𝑠𝜃)𝑑Ԧ𝑙′

𝑀𝑎𝑛𝑡𝑒𝑛ℎ𝑜 𝑜 𝑅 𝑑𝑒𝑛𝑡𝑟𝑜 𝑑𝑎 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙, 𝑝𝑜𝑖𝑠 𝑒𝑠𝑡𝑎𝑐𝑜𝑛𝑡𝑎 𝑠𝑒𝑟𝑣𝑒 𝑝𝑎𝑟𝑎 𝑞𝑢𝑎𝑙𝑞𝑢𝑒𝑟 𝑡𝑖𝑝𝑜 𝑑𝑒 𝑒𝑠𝑝𝑖𝑟𝑎,

𝑚𝑎𝑠 𝑠𝑒 𝑓𝑜𝑟 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 é 𝑐𝑙𝑎𝑟𝑜 𝑞𝑢𝑒 𝑅 é 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑒

Ԧ𝐴 =𝜇0𝑖

4𝜋

𝑛=0

∞

𝑟𝑛ර1

𝑅𝑛+1𝑃𝑛(𝑐𝑜𝑠𝜃)𝑑Ԧ𝑙′ 𝑟 < 𝑅

𝑪𝒂𝒎𝒑𝒐𝑴𝒂𝒈𝒏é𝒕𝒊𝒄𝒐 𝒅𝒆 𝒖𝒎𝒂 𝒆𝒔𝒑𝒊𝒓𝒂

q

x

y

z

i

Ԧ𝑟’

𝑑Ԧ𝑙′

Ԧ𝑟

𝑅𝜑

Ԧ𝐴 =𝜇0𝑖

4𝜋

1

𝑟ර𝑑Ԧ𝑙′ +

1

𝑟2ර𝑅𝑐𝑜𝑠𝜃𝑑Ԧ𝑙′ +

1

𝑟3ර𝑅2(

3

2𝑐𝑜𝑠2𝜃 −

1

2)𝑑Ԧ𝑙′ + ⋯

Monopoloර𝑑Ԧ𝑙′ = 01

𝑟2ර𝑅𝑐𝑜𝑠𝜃𝑑Ԧ𝑙′ Dipolo

1

𝑟3ර𝑅2(

3

2𝑐𝑜𝑠2𝜃 −

1

2)𝑑Ԧ𝑙′ Quadrupolo

𝑃𝑎𝑟𝑎 𝑟 ≫ 𝑅 Ԧ𝐴~𝜇0𝑖

4𝜋

1

𝑟2ර𝑅𝑐𝑜𝑠𝜃𝑑Ԧ𝑙′

ර𝑅𝑐𝑜𝑠𝜃𝑑Ԧ𝑙′ = ර Ƹ𝑟 ∙ 𝑅𝑑Ԧ𝑙′ = − Ƹ𝑟 × ර𝑑 Ԧ𝐴′ 𝑚𝑎𝑠 𝑖 ර𝑑 Ԧ𝐴′ = Ԧ𝜇 𝑚𝑜𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑑𝑖𝑝𝑜𝑙𝑜

Ԧ𝐴 =𝜇0𝑖

4𝜋

1

𝑟2− Ƹ𝑟 × ර𝑑 Ԧ𝐴′ =

𝜇04𝜋

1

𝑟2− Ƹ𝑟 × Ԧ𝜇 Ԧ𝐴 =

𝜇04𝜋

Ԧ𝜇 × Ƹ𝑟

𝑟2

𝑃𝑟𝑒𝑣𝑎𝑙𝑒𝑠𝑐𝑒 𝑜 𝑑𝑖𝑝𝑜𝑙𝑜

𝑟 > 𝑅

𝐵 = 𝛻 × Ԧ𝐴 =1

𝑟𝑠𝑒𝑛𝜃

𝜕𝑠𝑒𝑛𝜃𝐴𝜑

𝜕𝜃−𝜕𝐴𝜃𝜕𝜑

ො𝑎𝑟 +1

𝑟

1

𝑠𝑒𝑛𝜃

𝜕𝐴𝑟𝜕𝜑

−𝜕𝑟𝐴𝜑

𝜕𝑟ො𝑎𝜃 +

1

𝑟

𝜕𝑟𝐴𝜃𝜕𝑟

−𝜕𝐴𝑟𝜕𝜃

ො𝑎𝜑

Ԧ𝐴 =𝜇04𝜋

Ԧ𝜇 × Ƹ𝑟

𝑟2=𝜇04𝜋

𝜇

𝑟2𝑠𝑒𝑛𝜃 ො𝑎𝜑

q

x

y

z

i

Ԧ𝑟’

𝑑Ԧ𝑙′

Ԧ𝑟

𝑅𝜑

Ԧ𝜇

𝐵 =1

𝑟𝑠𝑒𝑛𝜃

𝜕𝑠𝑒𝑛𝜃𝐴𝜑

𝜕𝜃ො𝑎𝑟 −

1

𝑟

𝜕𝑟𝐴𝜑

𝜕𝑟ො𝑎𝜃

𝐵 =1

𝑟𝑠𝑒𝑛𝜃

𝜇04𝜋

𝜇

𝑟2𝜕𝑠𝑒𝑛2𝜃

𝜕𝜃ො𝑎𝑟 −

1

𝑟

𝜇04𝜋

𝜇𝑠𝑒𝑛𝜃𝜕

𝜕𝑟

1

𝑟ො𝑎𝜃

𝐵 =𝜇04𝜋

𝜇

𝑟32𝑐𝑜𝑠𝜃 ො𝑎𝑟 + 𝑠𝑒𝑛𝜃 ො𝑎𝜃

𝐵 =𝜇04𝜋

1

𝑟33 Ԧ𝜇. ො𝑎𝑟 ො𝑎𝑟 − Ԧ𝜇