Maxwell’s equation
Gauss’s Law
Charge <-> Electric field Flux
一个体积内的电荷 vs 这个体积表面的通量
\[ \nabla \cdot E = \frac{\rho}{\epsilon_0} \]
- 点电荷 \[ \rho = Q \delta(r) \]
\[ \int_V (\nabla \cdot E) dV = \int_S E \cdot dS = \int_V \frac{\rho}{\epsilon_0} dV = \frac{Q}{\epsilon_0} \] ???
\[ \int_S E \cdot dS = \int_V \frac{\rho}{\epsilon_0} dV \]
\[ LS= E \cdot R^2 dr d\theta \] \[ RS = \int \frac{Q \delta(r)}{\epsilon_0} dV \]
…
\[ dq = \ldots \] …
Q1. Find the electric potential distribution due to a sphere, a cylinder, and a film with uniform charge density \(\rho\)
- sphere \[ \int E \cdot dS = \frac{q}{\epsilon_0} \]
\[ \int E dS = E \int r^2 d\Omega \]
- 球外 \[ \frac{q}{\epsilon_0}=\frac{\frac{4}{3}\pi R^3 \rho}{\epsilon_0} \]
- 球内 \[ \frac{q}{\epsilon_0}=\frac{\frac{4}{3}\pi r^3 \rho}{\epsilon_0} \]
球壳，外面的电荷：
- 外面不会有 非径向，旋度
- 径向 积分为0
非均匀的带电体，内部画个圆的圆面上的\(\int E dS\),\(E\) 提不出来
- cylinder
- film ….
Surface Charges and Discontinuous
\[ \hat{n} \times (E_{+}-E_-) = 0 \]
\[ E_+ = E_{||} + E_{\perp} \] means: \(E_{||}\) 在表面是0
Electrostatic Equilibrium
Can we have a stable electrostatic system by only using a set of charges?
Argument:
背景场产生的电场指向中心，使得稳定，所以圈内有电荷，矛盾
Thompson, Putting
- 静电荷不能稳定
- 电子动要辐射
所以就说负电荷粘在一个正电荷的 pudding,至于正电荷如何稳定不管