Coulombs Law

Force between two charges

E=F₀/q₀

E=1/(ϵ₀pi4)*q/(r^2)ȓ

(note: E is directional) 

p=q*d

Ƭ=pEsin(ɸ)

Ƭ=p x E

U=-p . E

1/(4piϵ₀)*qx/(x²+a²)^3/2

When a ring charge is taken to its infinite limit, it resembles a point charge. 

E=1/(4piϵ₀)*q/(x√[x²+a²])

when infinite 

E=ƛ/(2piϵ₀r)

E=σ/(2ϵ₀)*(1-1/(√[r²-x²]+1))

if disk is very large, or point is very close,

Acts as an infinite sheet of charge

E=σ/(2ϵ₀)

Electric Flux=∫Ecos(ɸ)dA

=∫E⟂dA = ∫E . dA

Electric Flux=∮Ecos(ɸ)dA

=∮E⟂dA = ∮E . dA = q_enc/ϵ₀

Outside sphere = point charge

Inside = 0

outside cylinder = infinite wire

inside cylinder = 0

Two oppositely charged conducting plates with surface charge densities -σ and +σ

or 

Charged Conductor

U=qq₀/(4piϵ₀r) 

U=q₀/(4piϵ₀)(q₁/r₁+q₂/r₂+q₃/r₃+q₄/r₄....)

V=U/q₀

=q/(4piϵ₀r) 

V=1/(4piϵ₀)(q₁/r₁+q₂/r₂+q₃/r₃+q₄/r₄....)

V=1/(4pi)∫dq/r

V_a-V_b=∫(from a to b)E . dl =∫(from a to b)Ecos(ɸ)dl

Capacitance

C=KC₀

C₀ being the original capacitance

Kϵ₀=ϵ (permittivity) 

KC₀=Kϵ₀A/d

∮KE . dA = Q_encl-free/ϵ₀

J=nqv_d

(note: J and v are directional)

ρ=E/J

ρ(T)=ρ₀[1+ɑ(T-T₀)]

Ohm's Law 

(and resistors)

V=IR

R=ρL/A

i=dq/dt=ℇ/R*e^-t/RC

=I₀e^-t/RC

F=qv x B

(Note: F v and B are directional)

Magnetic Flux =∫B⟂dA

=∫Bcos(ɸ)dA

=∫B . dA

∮B . dA=0

F=Il x B

dF=I dl x B

(Note: F, l, and B are directional)

Ƭ=IBAsin(ɸ)

µ=IA

(A is directional in this instance)

Ƭ=µ x B

U= -µ . B = -µBcosɸ

µ=IA

Magnetic Field of Current Carrying Conductor

Biot Savart

a is radius

B_x=(µ₀Ia²)/(2(x²+a²)^3/2)

when x is 0 but there are N multiple loops:

B_x=µ₀NI/(2a)

ℇ=vBL

=∮(v x b) . dl

ℇ₂=-Mdi₁/dt 

and 

ℇ₁=-Mdi₂/dt 

M=N₂(electric flux 2)/i₁=N₁(electric flux 1)/i₂

ℇ=-Ldi/dt

L=N(magnetic flux)/i

U=LI²/2