Physics-Intro to Electrostatics

k = 1/(4 π ε ₀ ) = 8.99 x 10 ⁹ N m ² / C ² 

k is the proportionality constant in Coulomb's law, and is sometimes known as Coulomb's constant.

ε ₀ = 8.85 x 10 ^-12 C ² / Nm ² 
ε ₀ is a constant known as the permittivity of free space, which is the absolute value of the dielectric permittivity of a vacuum.

F ₁₂ = (k Q ₁ Q ₂ )/ r ² ₁₂
Coulomb force law, inverse square law between objects 1 and 2 with a given charge a s a function of distance squared. Can be both attractive and repulsive in nature.

E = Q r / (4 π ε ₀ r ² )
Electric field due to a point charge, written with ε ₀ as opposed to using constant k.

Dipole field

E = [3 ( p . r ) r - p] / (4 π ε ₀ r ³ ) - [p δ ³ (r)] / (3 ε ₀ )

In one direction along or perpendicular to dipole axis

E = p / (2 π ε ₀ x ³ )

Energy and Torque on a dipole in an external E-field

U = - p . E , τ = p x E

Electric field of infinite line of charge:
E = λ / (2 π ε ₀ r)
Where λ is defined as the charge per unit length.

Electric field of infinite charged plane:
E = σ / (2 ε ₀ )
Where σ is defined as the charge per unit area.

Electric field of ring along axis:
E = Q x / [ 2 π ε ₀ r (x ² + a ² ) ^3/2 ]

Electric field of disk along axis
E = σ / 2 ε ₀ [ 1 - z / (z ² + R ² ) ^1/2 ]

Electric flux:
Φ = E . A ;
dΦ = E . dA ; Φ = ∫ E . dA

Gauss's Law:

§ E . dA = Q _enc / ε ₀

Potential energy

U _ab = U _b - U _a = -W _ab = - ∫ F . dl ;
V _ab = V _b - V _a = - ∫ E . dl ; V = k Q / r ; V = k ∫ dq / r ; E _l = - dV / dl
E _x = - dV / dx
U ₁₂ = k Q ₁ Q ₂ / r ₁₂

Capacitors

Q = C V

Parallel plate

Q = ε ₀ A / d
U = Q ² / 2 C
C = K C ₀ 
ε = K ε ₀ 
C = C ₁ + C ₂ 
1/C = 1/C2 + 1/C3

Ohm's law

I = V/R = Δ Q / Δ t

Resistivity

ρ = ρ ₀ [ 1 + α (T - T ₀ )]

Power

P = VI = V ² / R = I ² R
The basic relation P = VI was transformed to the other two states by using Ohm's law.

Alternating Current

V = V ₀ sin ω t
P = V ² ₀ sin ² (ω t) / R
P _avg = V ² ₀ / (2 R)

Current density

j = I/A = n e v _drift = E / ρ
e = -1.6 x 10 ^-19 C