Electrostatics and Magnetostatics Cheat Sheet Formulas PDF

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EEC 130A : Formula Sheet Up to Midterm 2 Updated: Mar. 7th 2012 1 Electrostatics Z 0 1 ˆ 0 ρl dl E= R Force on a point charge q inside a static elec- 4π l0 R02 tric field Electric field produced by an infinite sheet F = qE of charge Gauss’s law ρs E = zˆ I 2 D · dS = Q or ∇·D=ρ S Electric field pr...

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EEC 130A : Formula Sheet Up to Midterm 2 Updated: Mar. 7th 2012

1

Electrostatics

1 E= 4πǫ

Force on a point charge q inside a static electric field F = qE

Z

ˆ ′ ρl dl R R′2 l′

Electric field produced by an infinite sheet of charge

Gauss’s law E = zˆ I

S

D · dS = Q

or

∇·D=ρ

∇×E=0

or

C

E=

E · dl = 0

q (R − Ri ) E= 4πǫ0 |R − Ri |3

E = −∇V

Electric field produced by a volume charge distribution Z

ˆ ′ ρv dV R R′2 V′

1 E= 4πǫ

ˆ ′ ρs ds R R′2 S′

or

V2 − V1 = −

Z

P2 P1

E · dl

Electric potential due to a point charge (with infinity chosen as the reference)

′

V =

Electric field produced by a surface charge distribution Z

Dr ρl D = ˆr = ˆr ǫ ǫ 2πǫr

Electric field - scalar potential relationship

Electric field produced by a point charge q in free space

1 E= 4πǫ

ρs 2ǫ

Electric field produced by an infinite line of charge

Electrostatic fields are conservative I

′

q 4πǫ0 |R − Ri |

Poisson’s equation ∇2 V = −

′

ρ ǫ

Constitutive relationship in dielectric materials D = ǫ0 E + P

Electric field produced by a line charge distribution 1

where P is the polarization.

Gauss’s law for magnetism I B · dS = 0 ∇ · B = 0 or

P = ǫ0 χe E

S

Electrostatic energy density Ampere’s law

1 we = ǫE 2 2

∇×H=J

Boundary conditions E1t = E2t

or

C

H · dl = I

Magnetic flux density — magnetic vector potential relationship

ˆ × (E1 − E2 ) = 0 n

or

I

B=∇×A D1n − D2n = ρs

or

ˆ · (D1 − D2 ) = ρs n

Magnetic potential produced by a current distribution

Ohm’s law J = σE

µ A= 4π

Conductivity σ = ρv µ

Z

V′

J dV ′ R′

Vector Poisson’s Equation

where µ stands for charge mobility. ∇2 A = −µJ

Joule’s law P =

2

Z

Magnetic field intensity produced by an infinitesimally small current element (BiotSavart law)

E · J dv

Magnetostatics dH =

ˆ I dl × R 4π R2

Force on a moving charge q inside a magnetic field F = qu × B

Magnetic field produced by an infinitely long wire of current in the z-direction

Force on an infinitesimally small current element Idl inside a magnetic field

ˆ I H=φ 2πr

dFm = Idl × B

Magnetic field produced by a circular loop of current in the φ-direction

Torque on a N -turn loop carrying current I inside a uniform magnetic field

H = zˆ

T=m×B

Ia2 2(a2 + z 2 )3/2

Constitutive relationship in magnetic materials

ˆ N IA. where m = n 2

4

Constants

B = µ0 H + µ0 M Free space permittivity Magnetization ǫ0 = 8.85 × 10−12 M = χm H

Free space permeability

Boundary conditions

B1n = B2n

or

H1t − H2t = Js

or

µ0 = 4π × 10−7 ˆ · (B1 − B2 ) = 0 n

ˆ × (H1 − H2 ) = Js n

Magnetostatic energy density 1 wm = µH 2 2

3

Useful Integrals √ dx = ln(x + x2 + c2 ) x2 + c2 Z 1 x dx = tan−1 2 2 x +c c c

Z

√

Z

dx


= 2 3/2

F/m

1 x √ 2 c x2 + c2

x2 + c Z √ x dx √ = x2 + c2 x2 + c2 Z 1 x dx = ln (x2 + c2 ) x2 + c2 2 Z x dx 1 = −√ 2 2 3/2 2 (x + c ) x + c2 Z 1 dx =− 2 (a + bx) b(a + bx)

3

H/m...