Electromagnetics II Formula Sheet for Mid1

Electromagnetics II

Formula Sheet

Cartesian Coordinates (x,y,z) A = Ax a x + Ay a y + Az a z ∂V ∂V ∂V ax + ay + az ∂x ∂y ∂z ∂Ay ∂Az ∂A ∇⋅A = x + + ∂x ∂y ∂z ∇V =

ax ∂ ∇×A = ∂x Ax

ay ∂ ∂y Ay

 Ax   cos φ     Ay  =  sin φ  A   0  z

− sin φ 0   Aρ    cos φ 0   Aφ  0 1   Az 

 Ax  sin θ cos φ     Ay  =  sin θ sin φ  A   cos θ  z

cos θ cos φ cos θ sin φ − sin θ

az  ∂A ∂Ay   ∂Ay ∂Ax  ∂  ∂Ax ∂Az  = z − − − ay +   ax +   az  ∂z  ∂ y ∂z  ∂x  ∂y   ∂z  ∂x Az

− sin φ   Ar    cos φ   Aθ  0   Aφ 

∂ 2V ∂ 2V ∂ 2V + + ∂x 2 ∂ y 2 ∂ z 2

∇ 2V =

Cylindrical Coordinates (ρ,φ,z)

A = Aρ a ρ + Aφ aφ + Az a z ∂V ∂V 1 ∂V ∇V = aρ + aφ + az ρ ∂φ ∂ρ ∂z 1 ∂ 1 ∂Aφ ∂Az ρ Aρ ) + ∇⋅A = + ( ρ ∂ρ ρ ∂φ ∂z aρ 1 ∂ ∇×A = ρ ∂ρ Aρ ∇ 2V =

ρ aφ

sin φ cos φ 0

0   Ax    0   Ay  1   Az 

az ∂Aρ   ∂Aρ ∂Az  1 ∂ ∂  1 ∂Az ∂Aφ  = − −  aρ +   aφ +  ( ρ Aφ ) −  az ∂z  ρ ∂φ ∂z  ∂ρ  ∂φ  ρ  ∂ρ  ∂z Az

∂ ∂φ ρ Aφ

1 ∂  ∂V ρ ρ ∂ρ  ∂ρ

 Aρ   cos φ     Aφ  =  − sin φ  A   0  z

 1 ∂ 2V ∂ 2V + 2 + 2 2 ∂z  ρ ∂φ

Spherical Coordinates (r,θ,φ) A = Ar a r + Aθ aθ + Aφ aφ ∇V =

1 ∂V 1 ∂V ∂V ar + aθ + aφ ∂r r ∂θ r sin θ ∂φ

∇⋅A =

1 ∂ 2 1 1 ∂Aφ ∂ r Ar ) + ( Aθ sin θ ) + ( 2 r ∂r r sin θ ∂θ r sin θ ∂φ

ar 1 ∂ ∇×A = 2 r sin θ ∂r Ar

raθ ∂ ∂θ rAθ

( r sin θ ) aφ ∂ ∂φ ( r sin θ ) Aφ

∂A   ∂A  1  ∂ 1  1 ∂Ar ∂ 1 ∂ − ( rAφ )  aθ +  ( rAθ ) − r  aφ Aφ sin θ ) − θ  a r +  (  ∂φ  r  ∂r ∂θ  r sin θ  ∂θ r  sin θ ∂φ ∂r  2 1 ∂  ∂V  1 1 ∂  ∂V  ∂V ∇ 2V = 2  r 2 + 2  sin θ + 2 2 r ∂r  ∂r  r sin θ ∂θ  ∂θ  r sin θ ∂φ 2  Ar   sin θ cos φ sin θ sin φ =

    Aθ  =  cos θ cos φ  Aφ   − sin φ  

cos θ sin φ cos φ

cos θ   Ax    − sin θ   Ay  0   Az 

Electromagnetics II

ε0 =

Formula Sheet

Coulomb’s Law due to N point charges: Electric Field Intensity:

10 −9 F/m 36π

μ 0 = 4π × 10 −7 H/m

Evaluation of the E field due to an infinite line charge:

Electric Potential:

Relationship between E and V:

Evaluation of the E field due to an infinite sheet of charge: Energy density for n point charges:

Energy density in general:

Laplace’s Equation: Stoke’s theorem :

Current:

Divergence theorem :

Gauss’s Law:

The power density wP: (in watts/m3)

Convection current Conduction current Joule’s Law: density: density:

Biot-Savart’s Law:

Poisson’s Equation:

H due to an infinitely long line current:

H due to a straight filamentary conductor:

H due to an infinite sheet of current:

Vector magnetic potential:

Ampere’s circuit Law: Scalar magnetic potential: Magnetic flux density

μ 0 = 4π × 10

B = μ0 H

−7

Maxwell’s Equations for Static EM Fields:

∇ ⋅ D = ρv

 DdS =  ρ dv v

S

∇⋅B = 0

v

 BdS = 0 S

∇×E = 0

 Edl = 0 L

∇×H = J

 Hdl =  JdS L

S

H/m

H = −∇Vm

( if

J = 0) ,

∇ 2Vm = 0

( J = 0)

H due to a circular current loop:

H=

I ρ 2a z 2  ρ 2 + h 2 

B = ∇×A μ0 Idl A= ( line current ) 4π R L μ KdS A= 0 ( surface current ) S 4π R μ Jdv A= 0 ( volume current ) v 4π R

The magnetic flux through a surface: 32

Ψ =  BdS =  Adl S

L