3.4 Electric fields and potentials of point charges

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A First Course in String Theory

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(a) Show that for time-independent fields, the Maxwell equation \displaystyle{T_{0ij}=0} implies that \displaystyle{\partial_i E_j - \partial_j E_i = 0}. Explain why this condition is satisfied by the ansatz \displaystyle{\vec E = - \nabla \Phi}.

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Eq. (3.23):

\displaystyle{ \begin{aligned} T_{\mu \lambda \nu} &= \partial_\lambda F_{\mu \nu} + \partial_\mu F_{\nu \lambda} + \partial_\nu F_{\lambda \mu} \\ \end{aligned}}

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\displaystyle{ \begin{aligned} &\vec E \\ &= - \nabla \Phi \\ &= - \left( \partial_x, \partial_y, \partial_z \right) \Phi \\ \end{aligned}}

\displaystyle{ \begin{aligned} &\partial_i E_j - \partial_j E_i \\ &= \partial_i \partial_j \Phi - \partial_j \partial_i \Phi \\ \end{aligned}}

— Me@2023-03-18 11:08:24 AM

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2023.03.18 Saturday (c) All rights reserved by ACHK