Eletromagnetismo Newton Mansur

𝜀 = −𝑑

𝑑𝑡Φ𝐵

Lei de Faraday – Neumann - Lenz

Φ𝐵 = 𝐵 ∙ 𝑑𝐴

B

v ∆𝑉

+

-

I

I

I

I

𝜀 = − 𝑑𝑑𝑡

Φ𝐵

Lei de Faraday

Neumann

Lenz

B(t)

IInd IInd

𝑩𝑰𝒏𝒅

𝜀 = −𝑑

𝑑𝑡 𝐵 ∙ 𝑑𝐴

𝑬𝑰𝒏𝒅 𝑬𝑰𝒏𝒅

𝑬𝑰𝒏𝒅

𝑬𝑰𝒏𝒅 𝑬𝑰𝒏𝒅

𝑬𝑰𝒏𝒅

q

FB =BA cosq Φ𝐵 = 𝐵 ∙ 𝑑𝐴

q =wt

FB =BA coswt

e= -N 𝑑𝑑𝑡

FB(t)=-NBA (-wsenwt)

e= NBAw senwt

-1

-0,5

0

0,5

1

0 Pi/2 Pi 3Pi/2 2Pi

fem Fluxo

𝜀 = −𝑑

𝑑𝑡Φ𝐵

𝜀 = −𝑑

𝑑𝑡𝐿𝑖

Φ𝐵 = 𝐿𝑖

𝜀 = −𝐿𝑑𝑖

𝑑𝑡

L - Indutância

Indutor

𝐿 =Φ𝐵𝑖

𝑈 =1

2𝐿𝑖2 𝐿 =

2𝑈

𝑖2

𝑈 =1

2𝜇0 𝐵2𝑑𝑣 𝑈 =

1

2 𝐵.𝐻𝑑𝑣

Indutor 𝑈 =

1

2𝐿𝑖2

i

𝐵 =𝜇0𝑖

2𝜋𝑟

𝐵 <𝜇0𝑖

2𝜋𝑟

𝐵 <𝜇0𝑖

2𝜋𝑟

i

i i

i

𝐵 ∙ 𝑑𝑠 = 𝜇0𝑖

i

𝐵 ∙ 𝑑𝑠 = 𝜇0𝑖

𝐵 ∙ 𝑑𝑠 = 𝜇0𝑖

i i

i i

i i 𝐵 =𝜇0𝑖

2𝜋𝑟

i i

𝑞(𝑡) −𝑞(𝑡)

𝐸(𝑡)

𝐸 𝑡 =𝜍(𝑡)

𝜖0 =

𝑞(𝑡)

𝐴𝜖0 𝑞 𝑡 = 𝐴𝜖0𝐸(𝑡) 𝑖 =

𝑑𝑞

𝑑𝑡 = 𝐴𝜖0

𝑑𝐸(𝑡)

𝑑𝑡

𝑖 = 𝜖0𝑑𝐸(𝑡)𝐴

𝑑𝑡 𝑖 = 𝜖0

𝑑Φ𝐸𝑑𝑡

𝑖𝐷 = 𝜖0𝑑Φ𝐸𝑑𝑡

𝐶𝑜𝑟𝑟𝑒𝑛𝑡𝑒 𝑑𝑒 𝑑𝑒𝑠𝑙𝑜𝑐𝑎𝑚𝑒𝑛𝑡𝑜

𝐵 ∙ 𝑑𝑠 = 𝜇0(𝑖 + 𝑖𝐷)

𝐵 ∙ 𝑑𝑠 = 𝜇0(𝑖 + 𝜖0𝑑Φ𝐸𝑑𝑡

)

i i +

+

-

-

𝐵 ∙ 𝑑𝑠 = 𝜇0𝑖 𝐵 ∙ 𝑑𝑠 = 𝜇0𝑖𝐷

𝐵 ∙ 𝑑𝑠 = 𝜇0(𝑖 + 𝜖0𝑑Φ𝐸𝑑𝑡

) = 𝜇0 (𝑖 + 𝜖0𝑑

𝑑𝑡 𝐸 ∙ 𝑑𝐴 )

𝐿𝑒𝑖 𝑑𝑒 𝐺𝑎𝑢𝑠𝑠 𝐸 ∙ 𝑑𝐴 =𝑞𝑖𝑛𝑡𝜖0

𝐸𝑞𝑢𝑎çõ𝑒𝑠 𝑑𝑒 𝑀𝑎𝑥𝑤𝑒𝑙𝑙

𝐵 ∙ 𝑑𝐴 = 0

𝐿𝑒𝑖 𝑑𝑒 𝐴𝑚𝑝è𝑟𝑒

𝐿𝑒𝑖 𝑑𝑒 𝐹𝑎𝑟𝑎𝑑𝑎𝑦 𝐸 ∙ 𝑑𝑠 = −𝑑Φ𝐵𝑑𝑡

= −𝑑

𝑑𝑡 𝐵 ∙ 𝑑𝐴

𝐿𝑒𝑖 𝑑𝑒 𝐺𝑎𝑢𝑠𝑠

𝐸 ∙ 𝑑𝐴 = 𝛻 ∙ 𝐸𝑑𝑉 =𝑞𝑖𝑛𝑡𝜖0

=1

𝜖0 𝜌𝑑𝑉

𝐸𝑞𝑢𝑎çõ𝑒𝑠 𝑑𝑒 𝑀𝑎𝑥𝑤𝑒𝑙𝑙

𝛻 ∙ 𝐸 =𝜌

𝜖0 𝛻 ∙ 𝐵 = 0

𝐵 ∙ 𝑑𝑠 = 𝛻 × 𝐵 ∙ 𝑑𝐴 = 𝜇0 (𝑖𝐼𝑛𝑡 + 𝜖0𝑑

𝑑𝑡 𝐸 ∙ 𝑑𝐴 )

𝐸𝑞𝑢𝑎çõ𝑒𝑠 𝑑𝑒 𝑀𝑎𝑥𝑤𝑒𝑙𝑙

𝐿𝑒𝑖 𝑑𝑒 𝐴𝑚𝑝è𝑟𝑒

𝛻 × 𝐵 ∙ 𝑑𝐴 = 𝜇0 ( 𝐽 ∙ 𝑑𝐴 + 𝜖0𝑑

𝑑𝑡 𝐸 ∙ 𝑑𝐴 )

𝛻 × 𝐵 ∙ 𝑑𝐴 = (𝜇0𝐽 + 𝜇0𝜖0𝑑𝐸

𝑑𝑡) ∙ 𝑑𝐴

𝛻 × 𝐵 = 𝜇0𝐽 + 𝜇0𝜖0𝑑𝐸

𝑑𝑡

𝐸𝑞𝑢𝑎çõ𝑒𝑠 𝑑𝑒 𝑀𝑎𝑥𝑤𝑒𝑙𝑙

𝐿𝑒𝑖 𝑑𝑒 𝐹𝑎𝑟𝑎𝑑𝑎𝑦

𝐸 ∙ 𝑑𝑠 = −𝑑Φ𝐵𝑑𝑡

= −𝑑

𝑑𝑡 𝐵 ∙ 𝑑𝐴

𝛻 × 𝐸 ∙ 𝑑𝐴 = −𝑑𝐵

𝑑𝑡∙ 𝑑𝐴

𝛻 × 𝐸 = −𝑑𝐵

𝑑𝑡

𝐸𝑞𝑢𝑎çõ𝑒𝑠 𝑑𝑒 𝑀𝑎𝑥𝑤𝑒𝑙𝑙

𝛻 ∙ 𝐸 =𝜌

𝜖0 𝛻 ∙ 𝐵 = 0

𝛻 × 𝐵 = 𝜇0𝐽 + 𝜇0𝜖0𝑑𝐸

𝑑𝑡

𝛻 × 𝐸 = −𝑑𝐵

𝑑𝑡

𝐿𝑒𝑖 𝑑𝑒 𝐺𝑎𝑢𝑠𝑠

𝐿𝑒𝑖 𝑑𝑒 𝐴𝑚𝑝è𝑟𝑒

𝐿𝑒𝑖 𝑑𝑒 𝐹𝑎𝑟𝑎𝑑𝑎𝑦

𝐸𝑞𝑢𝑎çõ𝑒𝑠 𝑑𝑒 𝑀𝑎𝑥𝑤𝑒𝑙𝑙

𝛻 × 𝐸 = −𝑑𝐵

𝑑𝑡 𝐿𝑒𝑖 𝑑𝑒 𝐹𝑎𝑟𝑎𝑑𝑎𝑦

𝛻 × 𝛻 × 𝐸 = 𝛻 × −𝑑𝐵

𝑑𝑡= −

𝑑

𝑑𝑡𝛻 × 𝐵

𝛻2𝐸 − 𝛻 ∙ (𝛻 ∙ 𝐸) = −𝑑

𝑑𝑡𝛻 × 𝐵

𝛻 ∙ 𝐸 =𝜌

𝜖0 𝐿𝑒𝑖 𝑑𝑒 𝐺𝑎𝑢𝑠𝑠 𝛻 ∙ 𝐸 = 0 𝑝𝑎𝑟𝑎 𝜌 = 0

𝛻 × 𝐵 = 𝜇0𝐽 + 𝜇0𝜖0𝑑𝐸

𝑑𝑡 𝐿𝑒𝑖 𝑑𝑒 𝐴𝑚𝑝è𝑟𝑒

𝑝𝑎𝑟𝑎 𝐽 = 0 𝛻 × 𝐵 = 𝜇0𝜖0𝑑𝐸

𝑑𝑡

𝛻2𝐸 = −𝜇0𝜖0𝑑2𝐸

𝑑𝑡2

𝐸𝑚 1 𝑑𝑖𝑚𝑒𝑛𝑠ã𝑜

𝑑2𝐸𝑥(𝑧, 𝑡)

𝑑𝑧2= −𝜇0𝜖0

𝑑2𝐸𝑥(𝑥, 𝑡)

𝑑𝑡2= −

1

𝑣2𝑑2𝐸𝑥(𝑥, 𝑡)

𝑑𝑡2

𝑣 =1

𝜇0𝜖0= 2,998𝑥108

𝑚

𝑠= 𝑐

𝐸𝑞𝑢𝑎çõ𝑒𝑠 𝑑𝑒 𝑀𝑎𝑥𝑤𝑒𝑙𝑙

x

y

𝜇1

𝜇2

𝜇0

𝐵 = 𝐵0𝑎 𝑦

𝐻 =𝐵0𝜇2

𝑎 𝑦

𝐵 = 𝐵0𝑎 𝑦

𝐻 =𝐵0𝜇1

𝑎 𝑦

𝐵 = 𝐵0𝑎 𝑦

𝐻 =𝐵0𝜇0

𝑎 𝑦

𝐼1

𝐵1 Φ21 = 𝐵1 ∙ 𝑑𝑆 2 𝑀21 =

Φ21𝐼1

𝜀21 = −𝑁2𝑀21𝑑𝐼1𝑑𝑡

Φ12 = 𝐵2 ∙ 𝑑𝑆 1 𝑀12 =Φ12𝐼2

𝐵1 = 𝛻 × 𝐴 1 Φ21 = 𝛻 × 𝐴 1 ∙ 𝑑𝑆 2 = 𝐴 1. 𝑑𝑙 2

𝐴 =𝜇14𝜋

𝑗

𝑟𝑑𝑣 = 𝑁

𝜇1𝐼

4𝜋

𝑑𝑙

𝑟

𝜀12 = −𝑁1𝑀12𝑑𝐼2𝑑𝑡

Φ21 = 𝑁1𝑁2𝜇1𝐼14𝜋

𝑑𝑙 1𝑟

. 𝑑𝑙 2 𝑀21 = 𝑁1𝑁2𝜇14𝜋

𝑑𝑙 1𝑟

. 𝑑𝑙 2 = 𝑀12

𝑀21 = 𝑁1𝑁2𝜇14𝜋

𝑑𝑙 1𝑟

. 𝑑𝑙 2 = 𝑀12

𝐼 𝐵

𝑁𝐼 = 𝐻. 𝑑𝑙 = 𝐻𝑙 =𝐵

𝜇𝑙 = 𝐵𝑆

𝑙

𝜇𝑆 = Ψ

𝑙

𝜇𝑆

𝜀 = 𝐼𝑅 = 𝐼𝑙

𝜍𝑆 ℱ = Ψ

𝑙

𝜇𝑆

ℱ = 𝑁𝐼 → 𝐹𝑜𝑟ç𝑎 𝑀𝑎𝑔𝑛𝑒𝑡𝑜𝑚𝑜𝑡𝑟𝑖𝑧

𝐼 = 𝑗 . 𝑑𝑆 Ψ = 𝑗 . 𝑑𝑆

Ψ = 𝑗 . 𝑑𝑆 → 𝐹𝑙𝑢𝑥𝑜 𝑀𝑎𝑔𝑛é𝑡𝑖𝑐𝑜

ℛ =𝑙

𝜇𝑆→ 𝑅𝑒𝑙𝑢𝑡â𝑛𝑐𝑖𝑎 ℛ =

ℱ

Ψ

𝐼 𝐵

ℛ =ℱ

Ψ

𝑅 ℰ

𝑅 =ℰ

𝐼

𝐼 𝐵

𝑙2

ℛ1 =𝑙1𝜇1𝑆

ℛ2 =𝑙2𝜇0𝑆

ℛ𝑇 = ℛ1 + ℛ2

Ψ =ℱ

ℛ𝑇

𝑅2

ℰ 𝑅1

𝐵

𝑙2

ℛ1 =𝑙1𝜇1𝑆

ℛ2 =𝑙2𝜇0𝑆

ℛ𝑇 =ℛ3ℛ4ℛ3+ℛ4

+ ℛ1 + ℛ2 Ψ =ℱ

ℛ𝑇

𝑅4

ℰ

𝑅3 𝑅2

𝑅1

ℛ3 =𝑙3𝜇1𝑆

ℛ4 =𝑙4𝜇0𝑆

ℇ

ℰ = 𝑅𝑖 +𝑑Ψ

𝑑𝑡 ℰ𝑖 = 𝑅𝑖2 + 𝑖

𝑑Ψ

𝑑𝑡 ℰ𝑖𝑑𝑡 = 𝑅𝑖2𝑑𝑡 + 𝑖𝑑Ψ

Ψ = 𝑁𝜙 = 𝑁𝐵𝑆 ℰ𝑖𝑑𝑡 = 𝑅𝑖2𝑑𝑡 + 𝑖𝑁𝑆𝑑𝐵 𝑖𝑁𝑆𝑑𝐵 → 𝑑1

2𝐿𝑖2 = 𝐿𝑖𝑑𝑖

𝐻 = 𝑛𝑖 =𝑁

𝑙𝑖 𝑁𝑖 = 𝐻𝑙 𝑑𝑊𝑚 = 𝐻𝑙𝑆𝑑𝐵 𝑊𝑚 = 𝑉 𝐻𝑑𝐵

𝐵𝑀𝑎𝑥

0

ℇ

𝑊𝑚 = 𝑉 𝐻𝑑𝐵 𝐵𝑀𝑎𝑥

0

𝐻 = 𝜇𝐵

𝐻

𝐵

𝐻𝑀𝑎𝑥

𝐵

𝐻𝑟

𝐵

𝐻

𝐵𝑀𝑎𝑥

𝐻

Os materiais ferromagnéticos duros têm ciclos de histerese com áreas grandes e dão

origem a maiores perdas por histerese, não sendo adequados para a construção de

máquinas eléctricas, em geral.

𝑊𝑚 =1

2𝐿𝑖2

Ψ = 𝑁𝜙 = 𝑁𝑁𝐼

𝑅𝑚 =

𝑁2

𝑅𝑚𝐼

𝐿 =𝑁2

𝑅𝑚

ℇ

𝐵(𝑡)

𝐼(𝑡)

𝐸𝐼𝑛𝑑

𝐵(𝑡)

𝐼(𝑡)

𝐸𝐼𝑛𝑑

- Materiais não duros com baixa histerese e baixa relutância

- Construído em lâminas isolados com materiais isolantes evitando corrente de Foucault

𝐼1

𝐵1 𝐴𝑐𝑜𝑝𝑙𝑎𝑚𝑒𝑛𝑡𝑜 𝑚𝑎𝑔𝑛é𝑡𝑖𝑐𝑜

𝑖1 → 𝜙1 = 𝜙11 + 𝜙12

𝑖2 → 𝜙2 = 𝜙22 + 𝜙21

Ψ11 = 𝑁1 𝜙11 + 𝜙12 = 𝑁1𝜙1

Ψ22 = 𝑁2 𝜙22 + 𝜙21 = 𝑁2𝜙2

Ψ1𝑡 = 𝑁1 𝜙1 ± 𝜙21 = Ψ11 ± Ψ21

Ψ2𝑡 = 𝑁2 𝜙2 ± 𝜙12 = Ψ22 ± Ψ12

Ψ1𝑡 = 𝐿11𝐼1 ± 𝑀21𝐼2

Ψ2𝑡 = 𝐿22𝐼2 ± 𝑀12𝐼1

𝐴𝑐𝑜𝑝𝑙𝑎𝑚𝑒𝑛𝑡𝑜 𝑚𝑎𝑔𝑛é𝑡𝑖𝑐𝑜

Ψ1𝑡 = 𝐿11𝐼1 ± 𝑀21𝐼2

Ψ2𝑡 = 𝐿22𝐼2 ± 𝑀12𝐼1

𝑀12 = 𝑀21 = 𝑀

ℇ1𝑡 = −𝑑Ψ1𝑡𝑑𝑡

= −𝐿11𝑑𝐼1𝑑𝑡

∓ 𝑀𝑑𝐼2𝑑𝑡

ℇ2𝑡 = −𝑑Ψ2𝑡𝑑𝑡

= −𝐿22𝑑𝐼2𝑑𝑡

∓ 𝑀𝑑𝐼1𝑑𝑡

ℇ1𝑡𝐼1 = −𝑑Ψ1𝑡𝑑𝑡

= −𝐿11𝐼1𝑑𝐼1𝑑𝑡

∓ 𝑀𝐼1𝑑𝐼2𝑑𝑡

ℇ2𝑡𝐼2 = −𝑑Ψ2𝑡𝑑𝑡

= −𝐿22𝐼2𝑑𝐼2𝑑𝑡

∓ 𝑀𝐼2𝑑𝐼1𝑑𝑡

𝑊𝑚𝑡 =1

2𝐿11𝐼1

2 +1

2𝐿22𝐼2

2 ± 𝑀21𝐼1𝐼2

𝐼1

𝐵1

𝐴𝑐𝑜𝑝𝑙𝑎𝑚𝑒𝑛𝑡𝑜 𝑚𝑎𝑔𝑛é𝑡𝑖𝑐𝑜

Ψ1𝑡 = 𝐿1𝐼1 + 𝑀𝐼2

Ψ2𝑡 = 𝐿2𝐼2 + 𝑀𝐼1

Ψ1𝑡Ψ2𝑡

=𝐿1 𝑀𝑀 𝐿2

𝐼1𝐼1

∆= 𝐿1𝐿2 − 𝑀2 = 𝐿1𝐿2 1 −

𝑀2

𝐿1𝐿2

𝑘 =𝑀2

𝐿1𝐿2 → 𝐶𝑜𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑒 𝑑𝑒 𝑎𝑐𝑜𝑝𝑙𝑎𝑚𝑒𝑛𝑡𝑜

𝑀 = 𝐿1𝐿2 → 𝑘 = 1 𝑎𝑐𝑜𝑝𝑙𝑎𝑚𝑒𝑛𝑡𝑜 𝑖𝑑𝑒𝑎𝑙

𝑀 = 0 → 𝑘 = 0 𝑠𝑖𝑠𝑡𝑒𝑚𝑎 𝑑𝑒𝑠𝑎𝑐𝑜𝑝𝑙𝑎𝑑𝑜

𝐼1

𝐵1

𝐴𝑐𝑜𝑝𝑙𝑎𝑚𝑒𝑛𝑡𝑜 𝑚𝑎𝑔𝑛é𝑡𝑖𝑐𝑜

ℇ1 = 𝐿1𝑑𝐼1𝑑𝑡

+ 𝑀𝑑𝐼2𝑑𝑡

ℇ2 = 𝐿2𝑑𝐼2𝑑𝑡

+ 𝑀𝑑𝐼1𝑑𝑡

𝐼1 = 𝑖1𝑒−𝑗𝜔𝑡

𝐼2 = 𝑖2𝑒−𝑗𝜔𝑡

ℇ1 = 𝑗𝜔 𝐿1𝐼1 + 𝑀𝐼2

ℇ2 = 𝑗𝜔 𝐿2𝐼2 + 𝑀𝐼1

𝐿1 ∝ 𝑁12

𝐿2 ∝ 𝑁22

𝐿1𝐿2

=𝑁1

2

𝑁22

𝑘2 =𝑀2

𝐿1𝐿2 𝐿1𝐿2 =

𝑀2

𝑘2

𝐿1𝑀

=𝑁1𝑁2𝑘

𝐿2𝑀

=𝑁2𝑁1𝑘

𝐿1𝑀

ℇ2 = 𝑗𝜔𝑀𝐿1𝐿2𝑀2

𝐼2 +𝐿1𝑀

𝐼1 ℇ1 = 𝑗𝜔𝑀𝐿1𝑀

𝐼1 + 𝐼2

𝐼1

𝐵1

𝐴𝑐𝑜𝑝𝑙𝑎𝑚𝑒𝑛𝑡𝑜 𝑚𝑎𝑔𝑛é𝑡𝑖𝑐𝑜

ℇ1 =𝐿1𝑀

ℇ2 + 𝑗𝜔𝑀 1 −𝐿1𝐿2𝑀2

𝐼2

ℇ1 =𝑁1𝑁2𝑘

ℇ2 + 𝑗𝜔𝑀 1 −1

𝑘2𝐼2

𝐼1 =1

𝑗𝜔𝑀ℇ2 −

𝑁2𝑁1𝑘

𝐼2

𝐴𝑐𝑜𝑝𝑙𝑎𝑚𝑒𝑛𝑡𝑜 𝑖𝑑𝑒𝑎𝑙 → 𝑘 = 1

ℇ1 =𝑁1𝑁2

ℇ2

𝐼1 =1

𝑗𝜔𝑀ℇ2 −

𝑁2𝑁1

𝐼2

ℇ2 = 0 → 𝐶𝑢𝑟𝑡𝑜 𝑛𝑜 𝑠𝑒𝑐𝑢𝑛𝑑á𝑟𝑖𝑜 𝐼1 = −𝑁2𝑁1

𝐼2

1

𝑗𝜔𝑀ℇ2 → 𝐶𝑜𝑟𝑟𝑒𝑛𝑡𝑒 𝑑𝑒 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑧𝑎çã𝑜

𝐼1

𝐵1

𝐴𝑐𝑜𝑝𝑙𝑎𝑚𝑒𝑛𝑡𝑜 𝑚𝑎𝑔𝑛é𝑡𝑖𝑐𝑜

ℇ1 = 𝑗𝜔 𝐿1𝐼1 + 𝑀𝐼2

ℇ2 = 𝑅2𝐼2 + 𝑗𝜔 𝐿2𝐼2 + 𝑀𝐼1

ℇ2 = 𝑗𝜔 𝐿2𝐼2 + 𝑀𝐼1

ℇ1 = 𝑅1𝐼1 + 𝑗𝜔 𝐿1𝐼1 + 𝑀𝐼2

ℇ1ℇ2

=𝑅1 00 𝑅2

𝐼1𝐼1

+ 𝑗𝜔𝐿1 𝑀𝑀 𝐿2

𝐼1𝐼2

ℇ1ℇ2

=𝑅1 + 𝑗𝜔𝐿1 𝑗𝜔𝑀

𝑗𝜔𝑀 𝑅2 + 𝑗𝜔𝐿2

𝐼1𝐼2

𝐼1𝐼2

=𝑅1 + 𝑗𝜔𝐿1 𝑗𝜔𝑀

𝑗𝜔𝑀 𝑅2 + 𝑗𝜔𝐿2

−1 ℇ1ℇ2

𝐼1

𝐵1

𝛻2𝐸 = 𝜇0𝜖0𝜕2𝐸

𝜕𝑡2

𝐸𝑚 1 𝑑𝑖𝑚𝑒𝑛𝑠ã𝑜

𝜕2𝐸𝑥(𝑥, 𝑡)

𝜕𝑥2= 𝜇0𝜖0

𝜕2𝐸𝑥(𝑥, 𝑡)

𝜕𝑡2=

1

𝑣2𝜕2𝐸𝑥(𝑥, 𝑡)

𝜕𝑡2

𝑣 =1

𝜇0𝜖0= 2,998𝑥108

𝑚

𝑠= 𝑐

𝐸𝑞𝑢𝑎çõ𝑒𝑠 𝑑𝑒 𝑀𝑎𝑥𝑤𝑒𝑙𝑙

𝜕2𝐸𝑥(𝑥, 𝑡)

𝜕𝑥2=

1

𝑣2𝜕2𝐸𝑥(𝑥, 𝑡)

𝜕𝑡2 𝐸𝑥(𝑥, 𝑡) = 𝑋 𝑥 𝑇(𝑡)

1

𝑋

𝑑2𝑋

𝑑𝑥2=

1

𝑇𝑣2𝑑2𝑇

𝑑𝑡2= −𝛽2 𝑇 𝑡 = 𝑇0𝑒

−𝑗𝛽𝑣𝑡

𝑋 𝑥 = 𝑋0𝑒±𝑗𝛽𝑥

𝐸𝑥(𝑥, 𝑡) = 𝐸0𝑒−𝑗𝛽(±𝑥+𝑣𝑡) 𝛽 =

2𝜋

𝜆

𝛽𝑣 = 𝜔 =2𝜋

𝑇 𝐸 (𝑟 , 𝑡) = 𝐸0𝑒

−𝑗 (±𝛽.𝑟 +𝜔𝑡)

𝐸 (𝑟 , 𝑡) = 𝐸0𝑒−𝑗 (𝑘.𝑟 ±𝜔𝑡)

𝛽 =2𝜋

𝜆 𝜔 =

2𝜋

𝑇

𝑣 =𝜔

𝛽

𝐸𝑥(𝑥, 𝑡) = 𝐸0𝑒−𝑗 (±𝛽𝑥+𝜔𝑡)

𝛻 ∙ 𝐸 =𝜌

𝜖0 𝛻 ∙ 𝐵 = 0

𝛻 × 𝐻 = 𝐽 + 𝜖0𝜕𝐸

𝜕𝑡

𝛻 × 𝐸 = −𝜕𝐵

𝜕𝑡

𝑆𝑒𝑚 𝑐𝑎𝑟𝑔𝑎 𝑙𝑖𝑣𝑟𝑒 𝑒 𝐸 (𝑟 , 𝑡) = 𝐸0 (𝑟 )𝑒𝑗𝜔𝑡

𝛻 ∙ 𝐸 =𝜌

𝜖0 𝛻 ∙ 𝐵 = 0

𝛻 × 𝐻 = 𝜍 + 𝑗𝜔𝜖 𝐸

𝛻 × 𝐸 = −𝑗𝜔𝜇𝐻

𝐽 = 𝜍𝐸

𝐶𝑜𝑛𝑑𝑢𝑡𝑜𝑟 𝑐𝑜𝑚 𝑐𝑜𝑛𝑑𝑢𝑡𝑖𝑣𝑖𝑑𝑎𝑑𝑒 𝑏𝑎𝑖𝑥𝑎 (𝑑𝑖𝑒𝑙é𝑡𝑟𝑖𝑐𝑜 𝑐𝑜𝑚 𝑝𝑒𝑟𝑑𝑎)

𝐶𝑜𝑛𝑑𝑢𝑡𝑜𝑟 𝑐𝑜𝑚 𝑐𝑜𝑛𝑑𝑢𝑡𝑖𝑣𝑖𝑑𝑎𝑑𝑒 𝑏𝑎𝑖𝑥𝑎 (𝑑𝑖𝑒𝑙é𝑡𝑟𝑖𝑐𝑜 𝑐𝑜𝑚 𝑝𝑒𝑟𝑑𝑎)

𝜕2𝐸𝑥(𝑥, 𝑡)

𝜕𝑥2= 𝑗𝜔𝜇 𝜍 + 𝑗𝜔𝜖 𝐸𝑥(𝑥, 𝑡) 𝛻

2𝐸 = 𝑗𝜔𝜇 𝜍 + 𝑗𝜔𝜖 𝐸

𝐸𝑥(𝑥, 𝑡) = 𝐸0𝑒−𝑗 (±𝛾𝑥+𝜔𝑡) 𝛾2 = 𝑗𝜔𝜇 𝜍 + 𝑗𝜔𝜖

𝐸𝑥(𝑥, 𝑡) = 𝐸0𝑒−𝛼𝑥cos (𝜔𝑡 ± 𝛽𝑥)

𝛼 = 𝜔𝜇𝜖

21 +

𝜍

𝜔𝜖

2

− 1

𝛽 = 𝜔𝜇𝜖

21 +

𝜍

𝜔𝜖

2

+ 1

?

𝐶𝑜𝑛𝑑𝑢𝑡𝑜𝑟 𝑐𝑜𝑚 𝑐𝑜𝑛𝑑𝑢𝑡𝑖𝑣𝑖𝑑𝑎𝑑𝑒 𝑎𝑙𝑡𝑎 (𝑐𝑜𝑛𝑑𝑢𝑡𝑜𝑟 𝑞𝑢𝑎𝑠𝑒 𝑖𝑑𝑒𝑎𝑙)

𝛼 = 𝜔𝜇𝜖

21 +

𝜍

𝜔𝜖

2

− 1 =𝜇𝜔𝜍

2 𝛽 = 𝜔

𝜇𝜖

21 +

𝜍

𝜔𝜖

2

+ 1 =𝜇𝜔𝜍

2

𝛿 =2

𝜇𝜔𝜍

𝐸𝑥 𝑥, 𝑡 = 𝐸0𝑒−𝛼𝑥 cos 𝜔𝑡 ± 𝛽𝑥 = 𝐸0𝑒

−𝑥 𝛿 cos (𝜔𝑡 ± 𝛽𝑥)

skin depth

Profundidade de penetração

𝐷𝑖𝑒𝑙é𝑡𝑟𝑖𝑐𝑜 𝑠𝑒𝑚 𝑝𝑒𝑟𝑑𝑎

𝛼 = 𝜔𝜇𝜖

21 +

𝜍

𝜔𝜖

2

− 1 = 0 𝛽 = 𝜔𝜇𝜖

21 +

𝜍

𝜔𝜖

2

+ 1 = 𝜔 𝜇𝜖

𝜍 = 0

𝑀𝑒𝑖𝑜 𝑑𝑖𝑠𝑝𝑒𝑟𝑠𝑖𝑣𝑜

𝛽 = 𝜔 𝜇0𝜖(𝜔) 𝑣 =𝜔

𝛽=

1

𝜇0𝜖(𝜔)