91影视

Suppose$\mathbf{v}\mathbf{=}\mathbf{0}$ andlocalid="1654682194645" $\mathit{A}\mathbf{=}{\mathit{A}}_{\mathbf{0}}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{k}\mathbf{x}\mathbf{鈥�}\mathbf{蝇}\mathbf{t}\mathbf{)}\hat{\mathbf{y}}$, wherelocalid="1654682226085" ${\mathit{A}}_{\mathbf{0}\mathbf{}\mathbf{,}}\mathit{蝇}$, and kare constants. Find E and B, and check that they satisfy Maxwell鈥檚 equations in a vacuum. What condition must you impose localid="1654682236104" $\mathit{蝇}$on andk?

In Chapter 5, I showed that it is always possible to pick a vector potential whose divergence is zero (the Coulomb gauge). Show that it is always possible to choose$\mathbf{鈭�}\mathbf{.}\mathit{A}\mathbf{=}\mathbf{-}{\mathit{渭}}_{\mathbf{0}}{\mathit{蔚}}_{\mathbf{0}}\left(鈭�V/鈭�t\right)$, as required for the Lorenz gauge, assuming you know how to solve the inhomogeneous wave equation (Eq. 10.16). Is it always possible to pick$\mathbf{V}\mathbf{=}\mathbf{0}$ ? How about$\mathbf{A}\mathbf{}\mathbf{=}\mathbf{0}$ ?

The vector potential for a uniform magnetostatic field is$\mathit{A}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\left(r脳B\right)$ (Prob. 5.25). Show that $\frac{\mathbf{d}\mathbf{A}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\left(v脳B\right)$, in this case, and confirm that Eq. 10.20 yields the correct equation of motion.

Derive Eq. 10.23. [Hint: Start by dotting v into Eq. 10.17.]