Điện từ học

Zero potential, nonzero field

• Midway between $+q$ and $-q$ the potential is 0, yet the field there is strong, pointing from $+q$ to $-q$.
• GRE electromagnetism tests $\mathbf E = -\nabla V$, Gauss's law 高斯定律 with symmetry, circuits, induction and EM waves.

Field is the slope of potential

• $\mathbf E = -\nabla V$: the field depends on how $V$ changes, not on its value. Where $V$ is constant over a region, $\mathbf E = 0$.
• Midway between two equal $+q$ charges: $E = 0$ but $V \ne 0$.

Gauss's law needs symmetry

• $\oint \mathbf E \cdot d\mathbf A = Q_{\text{enc}}/\varepsilon_0$ always holds, but it gives $E$ only when symmetry makes $E$ constant on the surface.
• Sphere: $E = \frac{Q}{4\pi\varepsilon_0 r^2}$ outside. Line: $E = \frac{\lambda}{2\pi\varepsilon_0 r}$. Plane: $E = \frac{\sigma}{2\varepsilon_0}$.
• Inside a uniformly charged shell, $E = 0$ and $V$ is constant — equal to its surface value, not zero.

Circuits and RC decay

• Kirchhoff: currents into a node sum to zero (charge); voltages round a loop sum to zero (energy).
• Discharging capacitor: $V = V_0 e^{-t/RC}$, time constant 时间常数 $\tau = RC$. After one $\tau$, $V = V_0/e \approx 0.37V_0$.

Induction and waves

• $\mathcal E = -\dfrac{d\Phi_B}{dt}$; Lenz: the induced current opposes the change in flux.
• In vacuum, Maxwell's equations give waves with $E/B = c$, $\mathbf E \perp \mathbf B \perp$ direction of travel.

Recap