HSC Physics Revision Hub

8.1

Coulomb's Law and Electric Field

F = \frac{1}{4\pi\varepsilon_0}\,\frac{q_1 q_2}{r^2}

Force between point charges. $\varepsilon_0 = 8.85\times 10^{-12}$ C²/(N·m²) is the permittivity of free space; like charges repel, unlike attract.

Electric field

\vec{E} = \frac{\vec{F}}{q_0} \qquad E = \frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}

Field is force per unit positive test charge; units N/C = V/m. Fields superpose vectorially.

Potential gradient

\vec{E} = -\frac{dV}{dr}\;\hat{r}

8.2

Gauss's Law and its Applications

Electric flux and Gauss's law

\oint \vec{E}\cdot d\vec{A} = \frac{q_{enclosed}}{\varepsilon_0}

The closed-surface flux of $\vec{E}$ equals the net charge enclosed over $\varepsilon_0$ . Distinct from electric potential $V$ - the flux $\Phi_E$ carries the dot symbol.

Applications (symmetric geometries)

Charged conducting shell/sphere: inside the conductor the field is zero ( $E = 0$ , charge resides on the surface); outside it behaves like a point charge at the centre, $E = kQ/r^2$ .

Uniformly charged solid sphere (radius R): inside, $E = \frac{Q r}{4\pi\varepsilon_0 R^3}$ (proportional to $r$ ); outside, point-like $E = kQ/r^2$ .

Infinite line/cylinder (charge/unit length $\lambda$ ): $E = \frac{\lambda}{2\pi\varepsilon_0 r}$ radial, for large length compared with distance.

Infinite plane sheet (surface charge $\sigma$ ): $E = \frac{\sigma}{2\varepsilon_0}$ uniform on each side. Between two parallel plates (opposite charge), $E = \sigma/\varepsilon_0$ .

8.3

Electric Potential

Point charge

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

Work done per unit charge in bringing a test charge from infinity. Units: volt (V) = J/C.

Electric dipole (far-field approximation)

V \approx \frac{1}{4\pi\varepsilon_0}\frac{p\cos\theta}{r^2} \qquad (r \gg d)

$p = qd$ is the dipole moment, $\theta$ the angle with the axis. Far-field fields: on the axis $E = 2kp/r^3$ , on the equator $E = kp/r^3$ . Dipole in a uniform field has torque $\tau = pE\sin\theta$ and potential energy $U = -pE\cos\theta$ .

8.4

Capacitance and Energy

Defining relation and parallel-plate capacitor

C = \frac{Q}{V} \qquad C = \frac{\varepsilon_0 A}{d}

$Q$ charge, $V$ potential difference, $A$ plate area and $d$ separation. Units: farad (F).

With a dielectric

C = \kappa C_0 = \frac{\kappa\varepsilon_0 A}{d}

A dielectric uniformly filling the capacitor multiplies capacitance by its relative permittivity $\kappa$ . (Full filling is assumed; the result differs if insertion is only partial.) Combinations: series $1/C_{eq} = \sum 1/C_i$ , parallel $C_{eq} = \sum C_i$ .

Energy stored in a capacitor

U = \frac12 QV = \frac12 C V^2 = \frac{Q^2}{2C}

Stored electrostatic energy, equivalently expressed. The energy density between the plates is $u = \frac12 \varepsilon_0 E^2$ (J/m³).