3.3 Electromagnetism in three dimensions, 2

A First Course in String Theory

(b) Repeat the analysis of three-dimensional electromagnetism starting with the Lorentz covariant formulation. Take $A^\mu = (\Phi, A^1, A^2)$ , examine $F_{\mu \nu}$ , the Maxwell equations (3.34), and the relativistic form of the force law derived in Problem 3.1.

~~~

$A^\mu = (\Phi, A^1, A^2)$

Eq. (3.20):

$F_{\mu \nu} = \begin{bmatrix} 0 & -E_x & -E_y & -E_z=0 \\ E_x & 0 & B_z & -B_y =0\\ E_y & -B_z & 0 & B_x = 0\\ E_z=0 & B_y=0 & -B_x=0 & 0\\ \end{bmatrix}$

Eq. (3.33):

$F^{\mu \nu} = \begin{bmatrix} 0 & E_x & E_y & E_z=0 \\ -E_x & 0 & B_z & -B_y =0\\ -E_y & -B_z & 0 & B_x = 0\\ -E_z=0 & B_y=0 & -B_x=0 & 0\\ \end{bmatrix}$

Eq. (3.34):

$\begin{aligned} \frac{\partial F^{\mu \nu}}{\partial x^\nu} &= \frac{1}{c} j^\mu \\ \end{aligned}$

$\begin{aligned} \frac{\partial F^{0 0}}{\partial x^0} + \frac{\partial F^{0 1}}{\partial x^1} + \frac{\partial F^{0 2}}{\partial x^2} + \frac{\partial F^{0 3}}{\partial x^3} &= \frac{1}{c} j^0 \\ \frac{\partial F^{1 0}}{\partial x^0} + \frac{\partial F^{1 1}}{\partial x^1} + \frac{\partial F^{1 2}}{\partial x^2} + \frac{\partial F^{1 3}}{\partial x^3} &= \frac{1}{c} j^1 \\ \frac{\partial F^{2 0}}{\partial x^0} + \frac{\partial F^{2 1}}{\partial x^1} + \frac{\partial F^{2 2}}{\partial x^2} + \frac{\partial F^{2 3}}{\partial x^3} &= \frac{1}{c} j^2 \\ \frac{\partial F^{3 0}}{\partial x^0} + \frac{\partial F^{3 1}}{\partial x^1} + \frac{\partial F^{3 2}}{\partial x^2} + \frac{\partial F^{3 3}}{\partial x^3} &= \frac{1}{c} j^3 \\ \end{aligned}$

$\begin{aligned} \frac{\partial F^{0 0}}{\partial x^0} + \frac{\partial F^{0 1}}{\partial x^1} + \frac{\partial F^{0 2}}{\partial x^2} &= \frac{1}{c} j^0 \\ \frac{\partial F^{1 0}}{\partial x^0} + \frac{\partial F^{1 1}}{\partial x^1} + \frac{\partial F^{1 2}}{\partial x^2} &= \frac{1}{c} j^1 \\ \frac{\partial F^{2 0}}{\partial x^0} + \frac{\partial F^{2 1}}{\partial x^1} + \frac{\partial F^{2 2}}{\partial x^2} &= \frac{1}{c} j^2 \\ \end{aligned}$

$\begin{aligned} \frac{\partial 0}{\partial x^0} + \frac{\partial E_x}{\partial x^1} + \frac{\partial E_y}{\partial x^2} &= \frac{1}{c} j^0 \\ \frac{\partial (-E_x)}{\partial x^0} + \frac{\partial 0}{\partial x^1} + \frac{\partial B_z}{\partial x^2} &= \frac{1}{c} j^1 \\ \frac{\partial (-E_y)}{\partial x^0} + \frac{\partial (-B_z)}{\partial x^1} + \frac{\partial 0}{\partial x^2} &= \frac{1}{c} j^2 \\ \end{aligned}$

$\begin{aligned} \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} &= \rho \\ - \frac{1}{c} \frac{\partial E_x}{\partial t} + \frac{\partial B_z}{\partial y} &= \frac{1}{c} j_x \\ - \frac{1}{c} \frac{\partial E_y}{\partial t} - \frac{\partial B_z}{\partial x} &= \frac{1}{c} j_y \\ \end{aligned}$

$\begin{aligned} \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} &= \rho \\ \frac{\partial B_z}{\partial y} &= \frac{1}{c} j_x + \frac{1}{c} \frac{\partial E_x}{\partial t} \\ - \frac{\partial B_z}{\partial x} &= \frac{1}{c} j_y + \frac{1}{c} \frac{\partial E_y}{\partial t} \\ \end{aligned}$

P. (3.1):

$\displaystyle{ \begin{aligned} \frac{d p_\mu}{ds} &= \frac{q}{c} F_{\mu \nu} \frac{d x^\nu}{ds} \\ \frac{d p_\mu}{ds} \left( \frac{ds}{dt} \right) &= \frac{q}{c} F_{\mu \nu} \frac{d x^\nu}{ds} \left( \frac{ds}{dt} \right) \\ \frac{d p_\mu}{dt} &= \frac{q}{c} F_{\mu \nu} \frac{d x^\nu}{dt} \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \frac{d p_0}{dt} &= \frac{q}{c} F_{0 0} \frac{d x^0}{dt} + \frac{q}{c} F_{0 1} \frac{d x^1}{dt} + \frac{q}{c} F_{0 2} \frac{d x^2}{dt} + \frac{q}{c} F_{0 3} \frac{d x^3}{dt} \\ \frac{d p_1}{dt} &= \frac{q}{c} F_{1 0} \frac{d x^0}{dt} + \frac{q}{c} F_{1 1} \frac{d x^1}{dt} + \frac{q}{c} F_{1 2} \frac{d x^2}{dt} + \frac{q}{c} F_{1 3} \frac{d x^3}{dt} \\ \frac{d p_2}{dt} &= \frac{q}{c} F_{2 0} \frac{d x^0}{dt} + \frac{q}{c} F_{2 1} \frac{d x^1}{dt} + \frac{q}{c} F_{2 2} \frac{d x^2}{dt} + \frac{q}{c} F_{2 3} \frac{d x^3}{dt} \\ \frac{d p_3}{dt} &= \frac{q}{c} F_{3 0} \frac{d x^0}{dt} + \frac{q}{c} F_{3 1} \frac{d x^1}{dt} + \frac{q}{c} F_{3 2} \frac{d x^2}{dt} + \frac{q}{c} F_{3 3} \frac{d x^3}{dt} \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \frac{d p_0}{dt} &= \frac{q}{c} F_{0 0} \frac{d x^0}{dt} + \frac{q}{c} F_{0 1} \frac{d x^1}{dt} + \frac{q}{c} F_{0 2} \frac{d x^2}{dt} \\ \frac{d p_1}{dt} &= \frac{q}{c} F_{1 0} \frac{d x^0}{dt} + \frac{q}{c} F_{1 1} \frac{d x^1}{dt} + \frac{q}{c} F_{1 2} \frac{d x^2}{dt} \\ \frac{d p_2}{dt} &= \frac{q}{c} F_{2 0} \frac{d x^0}{dt} + \frac{q}{c} F_{2 1} \frac{d x^1}{dt} + \frac{q}{c} F_{2 2} \frac{d x^2}{dt} \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \frac{d p_0}{dt} &= q (0) + \frac{q}{c} \left( - E_x \frac{d x}{dt} - E_y \frac{d y}{dt} \right) \\ \frac{d p_1}{dt} &= q E_x + \frac{q}{c} \left( (0) \frac{d x}{dt} + B_z \frac{d y}{dt} \right) \\ \frac{d p_2}{dt} &= q E_y + \frac{q}{c} \left( - B_z \frac{d x}{dt} + (0) \frac{d y}{dt} \right) \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \frac{d E}{dt} &= \vec v \cdot \vec F_E \\ \frac{d p_x}{dt} &= q \left( E_x + \frac{v_y}{c} B_z \right) \\ \frac{d p_y}{dt} &= q \left( E_y - \frac{v_x}{c} B_z \right) \\ \end{aligned}}$

— Me@2022-11-08 03:46:01 PM

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3.3 Electromagnetism in three dimensions, 2

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