Electrostatic Potential and Potential Energy — Notes

• Potential difference: $V_B - V_A = W_{AB}/q_0$, work per unit positive test charge moved (without acceleration) from A to B. Potential: work per unit charge to bring a test charge from infinity ($V_\infty = 0$). Unit volt = J C⁻¹; scalar.
• The electrostatic force is conservative: W is independent of path. A positive charge moves naturally from high to low potential.
• Equipotential surface: all points at the same V. No work along it; E is perpendicular to it; E points from higher to lower V; equipotentials are closer where E is larger; two never intersect. Point charge: concentric spheres; uniform field: parallel planes. Uniform field: $E = \Delta V/d$.

$W = -\int_\infty^{r}\dfrac{kqq_0}{x^2}dx = \dfrac{kqq_0}{r}$, so $V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r}$: positive near +q, negative near −q. Graphs: $V \propto 1/r$ while $E \propto 1/r^2$ (E falls faster); at any point $V/E = r$.

• System: $V = \dfrac{1}{4\pi\varepsilon_0}\sum_i \dfrac{q_i}{r_i}$ (algebraic sum, signs kept). V can be zero where E is not.
• Dipole ($\pm q$, separation $2a$, $p = 2qa$): axial $V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^2 - a^2} \approx \dfrac{kp}{r^2}$; equatorial $V = 0$; general point ($r \gg a$) $V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p\cos\theta}{r^2}$. Dipole potential falls as $1/r^2$.
• Dipole equipotentials: closed surfaces round each charge; the perpendicular bisector is the V = 0 equipotential.

• A charge $q$ at potential $V$: $U = qV$. 1 eV = $1.6 \times 10^{-19}$ J (energy of an electron through 1 V).
• Two charges: $U = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r_{12}}$ — the work to bring them from infinity; positive for like charges, negative for unlike (bound).
• Three charges: $U = k\left(\dfrac{q_1q_2}{r_{12}} + \dfrac{q_1q_3}{r_{13}} + \dfrac{q_2q_3}{r_{23}}\right)$ (each pair once). Work by an external agent = increase in U.

Work to rotate from θ₁ to θ₂: $W = \int pE\sin\theta\,d\theta = pE(\cos\theta_1 - \cos\theta_2)$. With $U(90°) = 0$: $U = -pE\cos\theta = -\vec p\cdot\vec E$.

• θ = 0°: $U = -pE$ (minimum, stable); θ = 90°: 0; θ = 180°: $+pE$ (maximum, unstable).
• Work to turn from 0° to 180°: $2pE$; from 0° to 90°: $pE$.