Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

3.3.7 Recap

Recap

Potential. Using $\nabla (1/ R) = - R / R^{3}$ , Coulomb's law becomes $E = - \nabla V$ with $V (P) = \frac{1}{4 π ε _{0}} \int \frac{d q ^{'}}{R} + V_{0} [V]$ , and for a point charge $V = q / (4 π ε_{0} R)$ . One scalar integral replaces three vector ones, and $E$ points "downhill" in $V$ .
Work and voltage. $W = q (V_{B} - V_{A})$ and $V_{B} - V_{A} = - \int_{A}^{B} E \cdot d l$ , independent of the path. Only differences are physical. Usually $V (\infty) = 0$ , so $V (P) = \int_{P}^{\infty} E \cdot d l$ is the work per unit charge to bring a charge from infinity. For sources extending to infinity, choose a finite reference.
First fundamental equation: $\nabla \times E = 0$ , $\oint_{C} E \cdot d l = 0$ . The electrostatic field is conservative, has no whirlpools, and its field lines never close. A field with non-zero curl cannot be electrostatic.
Equipotentials $V = const$ are perpendicular to the field lines, and $∣ E ∣ ≃ Δ V /Δ n$ is large where they crowd together. Conductors are equipotentials. For a point charge they are spheres. For a dipole, $V = p cos θ / (4 π ε_{0} r^{2})$ , with equipotentials $r^{2} = C cos θ$ and field lines $r = C sin^{2} θ$ .
Poisson and Laplace: $Δ V = - ρ_{v} / ε_{0}$ , and $Δ V = 0$ where there is no charge (then $V$ is the average of its surroundings and has no maxima or minima). With the conductor voltages as boundary values they give $V$ , then $E = - \nabla V$ and $ρ_{S} = D_{n}$ . One-dimensional solutions: $V = A x + B$ (plates), $A ln ρ + B$ (coaxial cylinders), $A / r + B$ (concentric spheres), $A ϕ + B$ (wedge).
Engineering in volts: $e V$ is the energy an electron gains (electron volt), and touch and step voltages are differences of $V = ρ I / (2 π r)$ around an earth electrode.