3.3 Electromagnetism in three dimensions, 2

A First Course in String Theory

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(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.

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

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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}

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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}

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Eq. (3.34):

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

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\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}

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\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}

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\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}

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\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}

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\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}

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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}}

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\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}}

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\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}}

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\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}}

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\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}}

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— Me@2022-11-08 03:46:01 PM

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