02 | October | 2022 | Physics Town

A First Course in String Theory

(b) Show explicitly that the Maxwell equations with sources emerge from (3.34).

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Eq. (3.34):

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

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

$\displaystyle{\begin{aligned} \frac{\partial F^{0 0}}{\partial x^0} + \frac{\partial F^{0 1}}{\partial x^1} + \frac{\partial F^{0 2}}{\partial x^2} + \frac{\partial F^{0 3}}{\partial x^3} &= \frac{1}{c} j^0 \\ \frac{\partial E_x}{\partial x^1} + \frac{\partial E_y}{\partial x^2} + \frac{\partial E_z}{\partial x^3} &= \rho \\ \nabla \cdot \mathbf{E} &= \rho \\ \\ \end{aligned}}$

$\displaystyle{\begin{aligned} \frac{\partial F^{1 0}}{\partial x^0} + \frac{\partial F^{1 1}}{\partial x^1} + \frac{\partial F^{1 2}}{\partial x^2} + \frac{\partial F^{1 3}}{\partial x^3} &= \frac{1}{c} j^1 \\ - \frac{\partial E_x}{\partial x^0} + \frac{\partial 0}{\partial x^1} + \frac{\partial B_z}{\partial x^2} - \frac{\partial B_y}{\partial x^3} &= \frac{1}{c} j^1 \\ \frac{\partial B_z}{\partial y} - \frac{\partial B_y}{\partial z} &= \frac{1}{c} j^1 + \frac{\partial E_x}{\partial x^0} \\ \end{aligned}}$

— Me@2022.10.02 11:49:59 AM

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3.2 Maxwell equations in four dimensions, 2