Conductors and Capacitors
Core concepts
In electrostatic equilibrium, the electric field inside a conductor is zero, excess charge resides on the surface, and the field just outside is perpendicular to the surface with magnitude (\sigma/\varepsilon_0). Charge concentrates at sharp points. A conductor shields its interior from external fields (Faraday cage).
A capacitor stores charge and energy. Capacitance (C = Q/\Delta V) depends only on geometry. Key results: parallel plates (C = \varepsilon_0 A/d); cylindrical and spherical forms follow from integrating the field. Stored energy is (U = \tfrac12 C(\Delta V)^2 = Q^2/(2C)). Dielectrics multiply capacitance by (\kappa) and reduce the field and voltage for fixed charge.
Traps
- Series vs. parallel: capacitors in parallel add directly ((C_{eq} = \sum C)), in series the reciprocals add. This is opposite to resistors.
- In series, charge is the same on each capacitor; in parallel, voltage is the same.
- Inserting a dielectric with the battery disconnected (Q fixed) raises V? No - it lowers V and E; with the battery connected (V fixed), Q increases. Read which quantity is held constant.
- Using (U = \tfrac12 QV) but forgetting the factor of one-half.
- Assuming field is zero inside a cavity containing charge; it is zero only if the cavity is empty.
Worked example
Two capacitors (C_1 = 2,\mu\text{F}) and (C_2 = 6,\mu\text{F}) in series across (12,\text{V}). Series equivalent: (1/C = 1/2 + 1/6 = 2/3), so (C_{eq} = 1.5,\mu\text{F}). Charge (Q = C_{eq}V = 18,\mu\text{C}) on each. Voltages: (V_1 = Q/C_1 = 9,\text{V}), (V_2 = 3,\text{V}), summing to 12 V. Rule of thumb: find the equivalent capacitance, then the common charge, then back out individual voltages.