EM 1- FORMULÁRIO

𝑘 = 1

4𝜋𝜀!≅ 9×10! 𝑚/𝐹

𝜀! = 8,854 × 10!!" 𝐹/𝑚 𝑒 = 1,602 ×10!!" 𝐶 𝑟! = 3,8 ×10!!" 𝑚 𝜌 = 𝑥! + 𝑦! , 𝜙 = 𝑡𝑔!! !

! , 𝑧 = 𝑧

𝑥 = 𝜌𝑐𝑜𝑠𝜙,𝑦 = 𝜌𝑠𝑒𝑛𝜙, 𝑧 = 𝑧 𝑎! = cos𝜙 𝑎! + sen𝜙 𝑎𝒚 𝑎𝝓 = − sen𝜙 𝑎! + cos𝜙 𝑎𝒚 𝑎𝒛 = 𝑎𝒛 𝑎! = cos𝜙 𝑎𝝆 − sen𝜙 𝑎𝝓 𝑎𝒚 = sen𝜙 𝑎𝝆 + cos𝜙 𝑎𝝓 𝑎𝒛 = 𝑎𝒛

𝑟 = 𝑥! + 𝑦! + 𝑧!, 𝜃 = 𝑡𝑔!! !!!!!

!

𝜙 = 𝑡𝑔!!𝑦𝑥

𝑥 = 𝑟 𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜙 , 𝑦 = 𝑟 𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜙 , 𝑧 = 𝑟 𝑐𝑜𝑠𝜃 𝑎! = sen𝜃 cos𝜙 𝑎𝒓 − cos𝜃 cos𝜙 𝑎𝜽 +−sen𝜃 𝑎𝝓 𝑎𝒚 = sen𝜃 sen𝜙 𝑎𝒓 + cos𝜃 sen𝜙 𝑎𝜽 + cos𝜃 𝑎𝝓 𝑎𝒛 = cos𝜃 𝑎𝒓−sen𝜃 𝑎𝝓 𝑎! = sen𝜃 cos𝜙 𝑎! + sen𝜃 sen𝜙 𝑎𝒚 + cos𝜃 𝑎𝒛 𝑎𝜽 = cos𝜃 cos𝜙 𝑎! + cos𝜃 sen𝜙 𝑎𝒚− sen𝜃 𝑎𝒛 𝑎𝝓 = − sen𝜙 𝑎! + cos𝜙 𝑎𝒚 𝑑𝑙 = 𝑑𝑥𝑎! + 𝑑𝑦𝑎𝒚 + 𝑑𝑧𝑎! 𝑑𝑆 = 𝑑𝑦 𝑑𝑧 𝑎! + 𝑑𝑥 𝑑𝑧 𝑎𝒚 + 𝑑𝑥 𝑑𝑦 𝑎𝒛 𝑑𝑣 = 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝑙 = 𝑑𝜌𝑎! + 𝜌𝑑𝜙𝑎! + 𝑑𝑧𝑎! 𝑑𝑆 = 𝜌 𝑑𝜙 𝑑𝑧 𝑎! + 𝑑𝜌 𝑑𝑧 𝑎! + 𝜌 𝑑𝜙 𝑑𝜌 𝑎𝒛

𝒅𝒗 = 𝝆 𝒅𝝆 𝒅𝝓 𝒅𝒛

𝑑𝑙 = 𝑑𝑟𝑎! + 𝑟 𝑑𝜃 𝑎! + 𝑟 sen𝜃 𝑑𝜙 𝑎! 𝑑𝑆 = 𝑟! sen𝜃 𝑑𝜃 𝑑𝜙 𝑎! + 𝑟 sen𝜃 𝑑𝑟 𝑑𝜙 𝑎𝜽 + 𝑟 𝑑𝑟 𝑑𝜃 𝑎𝝓 𝑑𝑣 = 𝑟! sen𝜃 𝑑𝑟 𝑑𝜃 𝑑𝜙

𝐴!𝐴!𝐴!

= cos𝜙 sen𝜙 0− sen𝜙 cos𝜙 0

0 0 1 𝐴!𝐴!𝐴!

𝐴!𝐴!𝐴!

= cos𝜙 − sen𝜙 0sen𝜙 cos𝜙 00 0 1

𝐴!𝐴!𝐴!

𝐴!𝐴!𝐴!

= sen𝜃 cos𝜙 sen𝜃 sen𝜙 cos𝜃cos𝜃 cos𝜙 cos𝜃 sen𝜙 − sen𝜃− sen𝜙 cos𝜙 0

𝐴!𝐴!𝐴!

𝐴!𝐴!𝐴!

= sen𝜃 cos𝜙 cos𝜃 cos𝜙 −sen𝜃sen𝜃 sen𝜙 cos𝜃 sen𝜙 cos𝜃cos𝜃 −sen𝜃 0

𝐴!𝐴!𝐴!

!E = Q

4πε0r2  ar  

!F12 =

14πε0

Q1Q2!R12

2 a12

!R12 =

!r2 −!r1

a12 =!R12!R12

!E =

!FQt

!E(!r ) = Q

4πε0!R2 aR =

Q4πε0

!r − !r '( )!r − !r ' 3

!E(!r ) = Qm

4πε0!r − !rm

2 amm=1

n

∑

ρv =dQdv           [C /m3]

Q = ρvvol.∫ !r '( )dv '

!E "r( ) = 1

4πε0ρvdv '"r − "r ' 2vol.

∫  aR  

!E "r( ) = 1

4πε0ρSdS '!r − !r ' 2S

∫  aR  

!E !r( ) = 1

4πε0ρldl '!r − !r ' 2C

∫  aR  

C → Caminho

dW = −Q

!E ⋅d!l

𝑉!" = 𝑉! − 𝑉!

∇×!E = 0

!E = − ∂V

∂ρaρ +

1ρ∂V∂φ

aφ +∂V∂z

az⎛

⎝⎜

⎞

⎠⎟  

!E = − ∂V

∂rar +

1r∂V∂θ

aθ +1

rsenθ∂V∂φ

aφ⎛

⎝⎜

⎞

⎠⎟    

V ≈

Q4πε0

d cosθr2

aR =!r − !r '!r − !r '

 

ρl =dQdl           [C /m]

Q = ρll∫ !r( )dl

V (!r ) = Q4πε0

1!r − !r '

Q = ρsS∫ !r( )dS

!E = ρl

2πε0ρaρ

!E = ρl

2πε0

x − x '( ) ax + y− y '( ) ayx − x '( )2 + y− y '( )2

!E = ρs

2ε0 aN

∇⋅!D =

1ρ

∂ ρDρ( )∂ρ

+1ρ

∂Dφ

∂φ+∂Dz

∂z

∇⋅!D =

1r2∂ r2Dr( )∂r

+1

r.senθ∂ senθ  Dθ( )

∂θ+

1r.senθ

∂Dφ

∂φ

ψ =Q    [C]!D =

Q4πr2

ar!D = ε0

!E

!D ⋅d!S =

S"∫ ρv dvV∫

ψ = dS!∫ ψ =

"D ⋅d!S

S!∫!D ⋅d!S =

S"∫ ∇ ⋅!Ddv

V∫∇⋅!D = ρv

∇⋅!D = lim

Δv→0

!D ⋅d!S

S"∫Δv

W = −Q!E ⋅d!l

inicial

final∫

VAB =Q4πε0

1rA−1rB

⎛

⎝⎜

⎞

⎠⎟

VA =Q4πε0

1rA

VAB =WQ= −

!E ⋅d!l

B

A∫     [V ]

V (!r ) = Qi

4πε01!r − !rii=1

n

∑

V (!r ) = 14πε0

ρv r '( )dv '!r − !r 'V∫

ρv r '( )dv '→ ρs r '( )dS '→ ρl r '( )dl '!E ⋅d!l"∫ = 0

∇× ∇V( ) = 0!E = −∇V          [V /m]

∇V =∂V∂r

ar +1r∂V∂θ

aθ +1

rsenθ∂V∂φ

aφ  

d!H =

Id!l '× aR4π!R2

!H =

12!K × an

!B = µ0

!H

µ0 = 4π ×10−7       [H /m]

ψm =!B ⋅d!S

S∫

ψm = 0

ψm =!B ⋅d!S

S"∫ = 0

∇⋅!B = 0

∇⋅!B = lim

Δv→0

!B ⋅d!S

S"∫Δv

!Fmag =Q

!v ×!B

!F =Q

!E + !v ×

!B( )

d!F = Id

!l ×!B

d!F =

!KdS( )×

!B

d!F =

!Jdv( )×

!B

!F = Id

!l ×!B"∫

𝜇! = 4𝜋 × 10!! 𝐻/𝑚 !H =

Id!l '× aR4π!R2

C"∫  

!H =

!KdS '( )× aR4π!R2

S∫

!E = Qd

4πε0r3 2cosθ ar + senθ aθ( )

!p =Q !d         [C.m]

V =!p ⋅ aR

4πε0!r − !r ' 2

aR =!r − !r '!r − !r '

WE =12 i  =  1

N

∑ Qi  Vi

WE =12

ρvV dvvol.∫

WE =12

!D ⋅!E( )dvvol.∫ =

ε02

!E

2dv

vol.∫

wE =dWE

dv=12!D ⋅!E             [J /m3]

I =!J ⋅d!S

S∫!J = ρv

!v  !J ⋅d!S

S"∫ = −∂Qi

∂tQi = ρv dvvol.∫

∇⋅!J = −∂ρv

∂t

∇⋅!J = lim

Δv→0

!J ⋅d!S

S"∫Δv!

F = −e!E

!vd = −µe

!E

!J = −ρeµe

!E

!J = −ρeµe

!E + ρhµh

!E  

!J =σ

!E

σ = −ρeµe

σ = −ρeµe + ρhµh

R = 1σLS

1ρv

∂ρv∂t

= −σε

ρv = ρv0e− t τ

τ =εσ!

E × aN S= 0

!D ⋅ aN S

= ρS

!H =

!Jdv '( )× aR4π!R2

V∫

!H =

I4πρ

cosα2 − cosα1( ) aφ

!H =

I2πρ

aφ !H ⋅d!l = I j

j∑

C"∫

!H ⋅d!l =

!J ⋅d!S

S∫∫

C"∫

 !H = nI  az !H ⋅d!l = ∇×

!H( ) ⋅d

!S

S∫∫

C"∫

∇×!H( ) ⋅ an = lim

Δs→0

!H ⋅d!l

C"∫ΔS

∇×!H =!J

∇×!H =

1ρ∂Hz

∂φ−∂Hφ

∂z⎛

⎝⎜

⎞

⎠⎟ aρ +

∂Hρ

∂z−∂Hz

∂ρ

⎛

⎝⎜

⎞

⎠⎟ aφ +

1ρ

∂ ρHφ( )∂ρ

−1ρ

∂Hρ

∂φ

⎛

⎝⎜⎜

⎞

⎠⎟⎟ az

∇×!H =

1rsenθ

∂ Hφsenθ( )∂θ

−∂Hθ

∂φ

⎛

⎝⎜⎜

⎞

⎠⎟⎟ ar +

1r

1senθ

∂Hr

∂φ−∂ rHφ( )∂r

⎛

⎝⎜⎜

⎞

⎠⎟⎟ aθ +

1r∂ rHθ( )∂r

−∂Hr

∂θ

⎛

⎝⎜

⎞

⎠⎟ aφ