Concepts and formulas

Electrostatics

The fundamental equations of electrostatics are linear equations,
∇·E = ρ/ε₀,  ∇×E = 0, (SI units).
The principle of superposition holds.

The electrostatic force on a particle with charge q at position r is F = qE(r).
∇×E = 0 <==> E = -∇Φ,  ∇²Φ = -ρ/ε₀.
Φ is the electrostatic potential.

Important formulas

The field at r due to a point charge at r':  E(r) = [1/(4πε₀)] q(r - r')/|r - r'|³.
The field of a charge distribution:
E(r) = [1/(4πε₀)][∫_v' dV' ρ(r')(r - r')/|r - r'|³ + ∫_A' dA' σ(r')(r - r')/|r - r'|³
+ ∫_l' dl' λ(r')(r - r')/|r - r'|³ + ∑_iq_i(r - r_i)/|r - r_i|³].
(We consider volume, surface, and line charge distributions and point charges.)

The potential at r due to a point charge at r':  Φ(r) = [1/(4πε₀)] q/|r - r'|.
The potential of a charge distribution:
Φ(r) = [1/(4πε₀)][∫_v' dV' ρ(r')/|r - r'| + ∫_A' dA' σ(r')/|r - r'|
+ ∫_l' dl' λ(r')/|r - r'| + ∑_iq_i/|r - r_i|].

Gauss' law:  ∫_{closed surface} E·dA = Q_inside/ε₀ .
In situations with enough symmetry, Gauss' law alone can be used to find E.

The electrostatic energy of a charge distribution:
U = ½∫Φ(r)dq(r) = ½[∫_vdV ρ(r)Φ(r) + ∫_AdA σ(r)Φ(r) + ∫_l dl λ(r)Φ(r) + ∑_iq_iΦ(r_i)],
or, for a continuous charge distribution, U = (ε₀/2)∫_{all space} E·E dV.

Dipoles

The dipole moment of a charge distribution centered at the origin is p = ∫_v' dV' ρ(r')r'.
The field of a dipole at the origin:  E(r) = [1/(4πε₀)](1/r³)[3(p·r)r/r² - p].
The potential of a dipole at the origin:  Φ(r) = [1/(4πε₀)](p·r)/r³.
The force on a dipole:  F = ∇(p·E).
The torque on a dipole:  τ = p×E.
The energy of a dipole in an external field:  U = -p·E.

Properties of conductors in electrostatics

• E = 0 inside.
• ρ = 0 inside.
• Any excess charge resides on the surface.
• E on the surface is perpendicular to the surface.
• E = (σ/ε₀)n just outside the surface.

Dielectrics

The polarization P = dp/dV is defined as the dipole moment per unit volume.
The total charge density is due to free and to bound (polarization) charges.
ρ = ρ_f + ρ_p,   σ = σ_f + σ_p,   ρ_p = -∇·P,   σ_p = P·n.
Definition: D = ε₀E + P  ---> ∇·D = ρ_f (Gauss' law for D).
 

For linear, isotropic, homogeneous (lih) dielectrics we have
P =  ε₀χ_eE, with χ_e constant.
D = ε₀(1 + χ_e)E = ε₀κ_eE = εE.
∇²Φ = -ρ_f/ε.

Energy in Dielectrics

The electrostatic energy stored in a charge distribution is given by 
U = (ε₀/2)∫_{all space} E·E dV.
In the presence of a dielectric, the total work done in assembling the free charges into a charge distribution is
W = ½∫_{all space} E·D dV,  
which becomes
W = (ε/2)∫_{all space} E·E dV.
in a linear, isotopic, homogeneous material .

W > U.  As you do work on the free charges against electrostatic forces, the electric field does work on the bound charges against non-electrostatic forces, thus lowering the total electrostatic potential energy stored in the system.  Some of the external work is stored as non-electrostatic potential energy.

Currents and Circuits

Current:   I = ∫j·dA  or  I = dQ/dt.
Resistance:   R = ΔV/I.
Resistance of a straight wire:   R = ρl/A.
Power:   P = IΔV = I²R = (ΔV)²/R.
Resistors in series:   R = R₁ + R₂ + R₃ + ... .
Parallel Resistors:   1/R = (1/R₁) + (1/R₂) + (1/R₃) + ... .

Kirchhoff's first rule :  (Junction rule)
At any junction point in a circuit where the current can divide, the sum of the currents into the junction must equal the sum of the currents out of the junction.  (This is a consequence of charge conservation.)

Kirchhoff's second rule :  (Loop rule)
When any closed circuit loop is traversed, the algebraic sum of the changes in the potential must equal zero.  (This is a consequence of conservation of energy.)