Electrostatic Potential and Capacitance

Cheatsheet Content

Capacitance of a Parallel Plate Capacitor Definition: $C = \frac{\epsilon_0 A}{d}$ $\epsilon_0$: permittivity of free space ($8.85 \times 10^{-12} \, \text{F/m}$) $A$: area of the plates $d$: distance between the plates With a dielectric material (dielectric constant $K$): $C_K = \frac{K \epsilon_0 A}{d} = K C$ Energy stored: $U = \frac{1}{2} C V^2 = \frac{Q^2}{2C} = \frac{1}{2} Q V$ Energy density: $u = \frac{1}{2} \epsilon_0 E^2$ Capacitors in Series and Parallel Series Connection: Equivalent Capacitance: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$ Charge is same across each capacitor: $Q_{total} = Q_1 = Q_2 = \dots$ Voltage divides: $V_{total} = V_1 + V_2 + \dots$ Parallel Connection: Equivalent Capacitance: $C_{eq} = C_1 + C_2 + \dots$ Voltage is same across each capacitor: $V_{total} = V_1 = V_2 = \dots$ Charge divides: $Q_{total} = Q_1 + Q_2 + \dots$ Dielectrics A dielectric slab inserted into a capacitor increases capacitance by a factor of $K$. Induced charge on dielectric: $Q_{in} = Q \left(1 - \frac{1}{K}\right)$ If a dielectric of thickness $t$ is inserted between plates of separation $d$: $C = \frac{\epsilon_0 A}{d - t + \frac{t}{K}}$ For a conducting slab ($K=\infty$) of thickness $t$: $C = \frac{\epsilon_0 A}{d - t}$ Electric Field and Potential Electric field due to a point charge $q$: $E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$ Electric potential due to a point charge $q$: $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$ Relationship between $E$ and $V$: $\vec{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$ Work done moving a charge $q$ in an electric field: $W = q (V_A - V_B)$ Electric field due to an infinite plane sheet of charge: $E = \frac{\sigma}{2\epsilon_0}$ Electric field inside a uniformly charged solid sphere (radius $R$, charge density $\rho$): $E = \frac{\rho r}{3\epsilon_0}$ for $r \le R$ Electric potential inside a charged spherical conducting shell (radius $R$, charge $Q$): $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{R}$ for $r \le R$ Common Potential and Energy Loss When two charged capacitors $C_1, V_1$ and $C_2, V_2$ are connected in parallel: Common Potential: $V_{common} = \frac{C_1 V_1 + C_2 V_2}{C_1 + C_2}$ Energy Loss: $\Delta U = \frac{1}{2} \frac{C_1 C_2}{C_1 + C_2} (V_1 - V_2)^2$ Electric Dipoles Dipole moment: $\vec{p} = q \vec{d}$ (from negative to positive charge) Torque on a dipole in an electric field: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential energy of a dipole in an electric field: $U = -\vec{p} \cdot \vec{E}$ Electric field on axial line (distance $r$ from center): $E_{axial} = \frac{1}{4\pi\epsilon_0} \frac{2p}{r^3}$ Electric field on equatorial plane (distance $r$ from center): $E_{equatorial} = \frac{1}{4\pi\epsilon_0} \frac{p}{r^3}$ Potential on equatorial plane: $V_{equatorial} = 0$