Show that Maxwell's equations given by 8.16 imply the wave equations [abla^{2} phi-frac{1}{c^{2}} frac{partial^{2} phi}{partial t^{2}}=-4 pi

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Show that Maxwell's equations given by 8.16 imply the wave equations

\[abla^{2} \phi-\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}=-4 \pi ho_{Q}\]

and

\[abla^{2} \mathbf{A}-\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}}=-\frac{4 \pi}{c} \mathbf{J}\]

To do this, you will need the vector identity \(\boldsymbol{abla} \times \boldsymbol{abla} \times \mathbf{A}=\boldsymbol{abla}(\boldsymbol{abla} \cdot \mathbf{A})-abla^{2} \mathbf{A}\). You will also need to fix the gauge freedom using 

image text in transcribed

The latter is allowed due to the gauge symmetry discussed in the text.

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Modern Classical Mechanics

ISBN: 9781108834971

1st Edition

Authors: T. M. Helliwell, V. V. Sahakian

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