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EletromagnetismoNewton Mansur
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Cargas em movimento
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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
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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
