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Formula Sheet — Electrostatics: Electric Charges and Potential

Electrostatics: Electric Charges and Potential

Constants and basic relations

Elementary charge: \(e = 1.6 \times 10^{-19}\,\mathrm{C}\)

Charge quantisation: \(q = ne\), where \(n\in\mathbb{Z}\)

Coulomb constant: \(k = \dfrac{1}{4\pi\varepsilon_0} \approx 9.0\times10^9\,\mathrm{N\,m^2\,C^{-2}}\)

Permittivity of free space: \(\varepsilon_0 \approx 8.85\times10^{-12}\,\mathrm{C^2\,N^{-1}\,m^{-2}}\)

Relative permittivity: \(\varepsilonr = \dfrac{\varepsilon}{\varepsilon0}\)

Charge density

Linear charge density: \(\lambda = \dfrac{dq}{dl}\), so \(dq = \lambda\,dl\)

Surface charge density: \(\sigma = \dfrac{dq}{dA}\), so \(dq = \sigma\,dA\)

Volume charge density: \(\rho = \dfrac{dq}{dV}\), so \(dq = \rho\,dV\)

For a uniformly charged object:

\(q = \lambda L\)

\(q = \sigma A\)

\(q = \rho V\)

Coulomb’s law

Magnitude of force between two point charges:

F = k\frac{|q1q2|}{r^2} = \frac{1}{4\pi\varepsilon0}\frac{|q1q_2|}{r^2}

Vector form:

\vec F{12} = \frac{1}{4\pi\varepsilon0}\frac{q1q2}{r^2}\hat r_{12}

In a medium of permittivity \(\varepsilon\):

F = \frac{1}{4\pi\varepsilon}\frac{|q1q2|}{r^2} = \frac{F0}{\varepsilonr}

Principle of superposition:

\vec F{\mathrm{net}} = \sumi \vec F_i

Electric field

Definition:

\vec E = \frac{\vec F}{q_0}

Field due to a point charge:

\vec E = \frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\hat r

Magnitude:

E = k\frac{|q|}{r^2}

Force on a charge in an electric field:

\vec F = q\vec E

For multiple charges:

\vec E{\mathrm{net}} = \sumi \vec E_i

Continuous charge distribution:

d\vec E = \frac{1}{4\pi\varepsilon_0}\frac{dq}{r^2}\hat r,

\qquad

\vec E = \frac{1}{4\pi\varepsilon_0}\int \frac{dq}{r^2}\hat r

Electric dipole

Dipole moment: \(\vec p = q\vec d\)

Magnitude: \(p = qd\)

Direction: from negative charge to positive charge

Electric field on the axial line, at distance \(r\) from the centre:

E{\mathrm{axial}} = \frac{1}{4\pi\varepsilon0}\frac{2pr}{(r^2-a^2)^2}

For a short dipole \((r\gg a)\):

E{\mathrm{axial}} \approx \frac{1}{4\pi\varepsilon0}\frac{2p}{r^3}

Electric field on the equatorial line, at distance \(r\) from the centre:

E{\mathrm{equatorial}} = \frac{1}{4\pi\varepsilon0}\frac{p}{(r^2+a^2)^{3/2}}

For a short dipole:

E{\mathrm{equatorial}} \approx \frac{1}{4\pi\varepsilon0}\frac{p}{r^3}

The equatorial field is opposite to \(\vec p\).

Torque on a dipole in a uniform electric field:

\vec\tau = \vec p\times\vec E,

\qquad

\tau = pE\sin\theta

Potential energy of a dipole:

U = -\vec p\cdot\vec E = -pE\cos\theta

Electric flux and Gauss’s law

Electric flux through a plane surface:

\Phi_E = \vec E\cdot\vec A = EA\cos\theta

For a non-uniform field:

\Phi_E = \int \vec E\cdot d\vec A

Gauss’s law:

\oint \vec E\cdot d\vec A = \frac{q{\mathrm{enclosed}}}{\varepsilon0}

Useful results:

Infinite line charge:

E = \frac{\lambda}{2\pi\varepsilon_0 r}

Infinite plane sheet of charge:

E = \frac{\sigma}{2\varepsilon_0}

Just outside a charged conductor:

E = \frac{\sigma}{\varepsilon_0}

Spherical shell or conducting sphere:

E =

\begin{cases}

0, & r<R\\

\dfrac{1}{4\pi\varepsilon_0}\dfrac{Q}{r^2}, & r\ge R

\end{cases}

Electric potential

Potential difference between two points:

VB - VA = -\int_A^B \vec E\cdot d\vec l

For a uniform electric field:

\Delta V = -Ed\cos\theta

Magnitude for displacement along the field:

|\Delta V| = Ed

Potential due to a point charge:

V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}

Potential due to several point charges:

V = \frac{1}{4\pi\varepsilon0}\sumi\frac{qi}{ri}

Potential due to a continuous charge distribution:

V = \frac{1}{4\pi\varepsilon_0}\int\frac{dq}{r}

Electric field–potential relation:

\vec E = -\nabla V

In one dimension:

E_x = -\frac{dV}{dx}

Potential of a dipole

Exact potential at a point:

V = \frac{1}{4\pi\varepsilon_0}\frac{p\cos\theta}{r^2}

This expression applies to a short dipole.

On the axial line \((\theta=0^\circ)\):

V = \frac{1}{4\pi\varepsilon_0}\frac{p}{r^2}

On the equatorial line \((\theta=90^\circ)\):

V = 0

Potential energy

Two point charges:

U = \frac{1}{4\pi\varepsilon0}\frac{q1q_2}{r}

System of point charges:

U = \frac{1}{4\pi\varepsilon0}\sum{i<j}\frac{qiqj}{r_{ij}}

Potential energy of a charge \(q\) at potential \(V\):

U = qV

Work done by the electric field:

W{\mathrm{field}} = -\Delta U = q(VA-V_B)

Work done by an external agent slowly:

W_{\mathrm{external}} = \Delta U

Conductors in electrostatic equilibrium

Electric field inside a conductor: \(E=0\)

Potential is constant throughout the conductor and on its surface.

Electric field at the surface is normal to the surface.

Just outside the surface: \(E=\dfrac{\sigma}{\varepsilon_0}\)

Excess charge resides on the outer surface.

Capacitance

Capacitance definition:

C = \frac{Q}{V}

Unit: \(1\,\mathrm{F}=1\,\mathrm{C\,V^{-1}}\)

Parallel-plate capacitor:

C = \frac{\varepsilon_0 A}{d}

With dielectric of relative permittivity \(K\):

C = \frac{K\varepsilon0 A}{d} = KC0

Energy stored:

U = \frac12QV = \frac12CV^2 = \frac{Q^2}{2C}

Energy density in an electric field:

u = \frac12\varepsilon_0E^2

With a dielectric:

u = \frac12\varepsilon E^2

Capacitors in parallel:

C{\mathrm{eq}} = C1+C_2+\cdots

Capacitors in series:

\frac{1}{C{\mathrm{eq}}} = \frac{1}{C1}+\frac{1}{C_2}+\cdots

For two capacitors in series:

C{\mathrm{eq}} = \frac{C1C2}{C1+C_2}

Quick conceptual checks

Electric field points in the direction of decreasing potential.

Equipotential surfaces are perpendicular to electric-field lines.

No work is done moving a charge along an equipotential surface.

Potential is a scalar, so potentials add algebraically.

Electric field is a vector, so fields must be added vectorially.

At electrostatic equilibrium, the electric field inside a conductor is zero, but the potential need not be zero; it is simply constant.

Common units

Charge: \(\mathrm{C}\)

Force: \(\mathrm{N}\)

Electric field: \(\mathrm{N\,C^{-1}}\) or \(\mathrm{V\,m^{-1}}\)

Potential: \(\mathrm{V}\)

Potential energy: \(\mathrm{J}\)

Dipole moment: \(\mathrm{C\,m}\)

Capacitance: \(\mathrm{F}\)

Electric flux: \(\mathrm{N\,m^2\,C^{-1}}\)

Sign reminder: positive charges produce positive potential and negative charges produce negative potential. For force and electric field, always determine the direction before assigning signs.