麦克斯韦大法好！！

Maxwell Dafa is good! !

先供上麦克斯韦方程 膜拜膜拜(　o=^•ェ•)o

$$
\nabla\cdot\vec{E} = \frac{\rho}{\varepsilon_0}
$$

$$
\nabla\cdot\vec{B} = 0
$$

$$
\nabla\times\vec{E} = -\frac{\partial B}{\partial t}
$$

$$
\nabla\times\vec{B} = \mu_0\left(\vec{J}+\varepsilon_0\frac{\partial E}{\partial t} \right)
$$

第一话 - 高斯生库伦

  • 假设空间中两点电荷$Q_{1}$,$Q_{2}$，相距d，欲求其相互作用的电场力。
  • 现以$Q_{1}$为圆心，$d$为半径做球。根据高斯law可知球面上的电通量只与球内电荷量有关，本例中为$\frac{Q_{1}}{\varepsilon_0}$。
  • 将上式中电通量除以求表面积可得电场强度$\frac{Q_{1}}{4\pi d^{2}\varepsilon_0}$
  • 场强乘以$Q_{2}$即可得库仑力$\frac{Q_{1}Q_{2}}{4\pi d^{2}\varepsilon_0}$
  • 令$k=\frac{1}{4\pi \varepsilon_0}$整理得
    $$F = \frac{k Q_{1}Q_{2}}{d^{2}}$$
    证毕

第二话 - 高斯金箍棒

  • 假设一均匀带电长度正无穷细杆，电荷密度为$\lambda$，求距其$d$处场强。
  • 绕杆画一个过待求点的圆柱，设高为$x$。
  • 由于杆长无限，圆柱两底面电场被抵消。
  • 圆柱侧面积为$2\pi dx$
  • 圆柱内电荷量为$\lambda x$
  • 引入高斯，得
    $$
    2\pi dxE = \frac{\lambda x}{\varepsilon_{0}}
    $$
    整理得
    $$
    E = \frac{\lambda}{2\pi d\varepsilon_{0}}
    $$
    证毕

高斯球球球

  • 首先要有一个带电小球，电量$Q$，想求其外部距其圆心$d$处场强。
  • 然后可列式
    $$
    4\pi d^2 E = \frac{Q}{\varepsilon_0}
    $$
    整理得
    $$
    E = \frac{Q}{4\pi \varepsilon_0 d^2}
    $$
    毕

高斯大面

  • 首先有一个均匀带电无穷面，带电面密度$\rho$，欲求距其$d$的点场强。
  • 以无穷面为中央横截面，做一个底面圆心为待求点的圆柱，半径为$r$。
  • 由于电场线皆平行，只有两个底面有电场线穿过。
  • 可列式
    $$
    2\pi r^2 E = \frac{\rho \pi r^2}{\varepsilon_0}
    $$
    整理得
    $$
    E = \frac{\rho}{2\varepsilon_0}
    $$
    完事

To Be Continued…

First offer to Maxwell’s equation, worship (　o=^•ェ•)o

$$
\nabla\cdot\vec{E} = \frac{\rho}{\varepsilon_0}
$$

$$
\nabla\cdot\vec{B} = 0
$$

$$
\nabla\times\vec{E} = -\frac{\partial B}{\partial t}
$$

$$
\nabla\times\vec{B} = \mu_0\left(\vec{J}+\varepsilon_0\frac{\partial E}{\partial t} \right)
$$

Chapter One-Gossian Cullen

  • Suppose two electric charges $Q_{1}$ and $Q_{2}$ in space are separated by d, and the electric field force they want to interact with.
  • Now take $Q_{1}$ as the center of the circle and $d$ as the radius to make the ball. According to the Gaussian law, the electric flux on the sphere is only related to the amount of charge in the sphere, which is $\frac{Q_{1}}{\varepsilon_0}$ in this example.
  • Divide the electric flux in the above formula by the surface area to get the electric field intensity $\frac{Q_{1}}{4\pi d^{2}\varepsilon_0}$
  • Field strength is multiplied by $Q_{2}$ to get Coulomb force $\frac{Q_{1}Q_{2}}{4\pi d^{2}\varepsilon_0}$
  • Let $k=\frac{1}{4\pi \varepsilon_0}$ to get
    $$F = \frac{k Q_{1}Q_{2}}{d^{2}}$$
    Completed

Chapter 2-Gauss Golden Cudgel

  • Assuming that a uniformly charged length is positively infinitely thin and the charge density is $\lambda$, find the field strength $d$ away from it.
  • Draw a cylinder around the rod and set the height to be $x$.
  • Due to the infinite length of the rod, the electric fields on the two bottom surfaces of the cylinder are cancelled.
  • The cylindrical side area is $2\pi dx$
  • The amount of charge in the cylinder is $\lambda x$
  • Introduce Gaussian, get
    $$
    2\pi dxE = \frac{\lambda x}{\varepsilon_{0}}
    $$
    Organized
    $$
    E = \frac{\lambda}{2\pi d\varepsilon_{0}}
    $$
    Completed

Gauss Ball Ball

  • First, there must be a charged ball with electricity $Q$, and I want to find the field strength at the distance of $d$ from the center of the ball.
  • Then columnable
    $$
    4\pi d^2 E = \frac{Q}{\varepsilon_0}
    $$
    Organized
    $$
    E = \frac{Q}{4\pi \varepsilon_0 d^2}
    $$
    complete

Gaussian Noodles

  • First, there is a uniformly charged infinite surface, the charged surface density is $\rho$, and the field strength of the point $d$ is desired.
  • Using the infinite plane as the central cross section, make a cylinder with the center of the bottom surface as the point to be found, and the radius is $r$.
  • Since the electric field lines are all parallel, only two bottom surfaces have electric field lines passing through.
  • Columnable
    $$
    2\pi r^2 E = \frac{\rho \pi r^2}{\varepsilon_0}
    $$
    Organized
    $$
    E = \frac{\rho}{2\varepsilon_0}
    $$
    Finished

To Be Continued…