Summary

This chapter developed WBCHSE Unit 1's second sub-topic, moving from the scalar description of the electrostatic field through to the capacitor as a practical charge-storage device. Electric potential, $V = W_{\infty\to P}/q_0$ (Section 2.2), and the exact relation connecting it to field, $E = -dV/dl$, or $V=Ed$ for a uniform field (Section 2.3), together let potential be found for a point charge, $V=kq/r$ (Section 2.4); a dipole, $V = kp\cos\theta/r^2$, exactly zero on the equatorial line (Section 2.5); and, by scalar superposition, any system of point charges, $V=k\sum q_i/r_i$ (Section 2.6). Equipotential surfaces (Section 2.7) -- always perpendicular to $\vec{E}$, never intersecting, more closely spaced where the field is stronger -- gave a geometric picture of the field: concentric spheres for a point charge, distorted closed surfaces for a dipole. Electrostatic potential energy, $U = kq_1q_2/r_{12}$ for two point charges, summed pairwise for more charges (Section 2.8), and $U(\theta) = -pE\cos\theta$ for a dipole in a uniform external field, minimum ($-pE$) when aligned and maximum ($+pE$) when anti-aligned (Section 2.9), described the energy cost of assembling or reorienting a charge configuration. Conductors were shown to have free charges that redistribute until the interior field is zero (making the whole conductor an equipotential body), while insulators have only bound charges, incapable of long-range motion (Section 2.10); an external field still polarizes a dielectric, by inducing or partially aligning molecular dipoles, producing bound surface charge and a reduced net field $E = E_0/K$ (Section 2.11). A capacitor, $C = Q/V$ in farads (Section 2.12), stores charge between two conductors; the parallel plate capacitor gives $C_0 = \epsilon_0A/d$ without a dielectric, $C = K\epsilon_0A/d$ filled completely, and $C = \epsilon_0A/[(d-t)+t/K]$ for a slab of thickness $t<d$ (Section 2.13); spherical capacitors give $C=4\pi\epsilon_0R$ for an isolated sphere and $C = 4\pi\epsilon_0\,ab/(b-a)$ for two concentric shells (Section 2.14). Combining capacitors (Section 2.15) follows rules opposite to resistors: in series (Section 2.15.1), same charge on each, $1/C_s = 1/C_1+1/C_2+\cdots$, equivalent capacitance smaller than any individual one, used to withstand a higher total voltage; in parallel (Section 2.15.2), same voltage across each, …