silverwhale said:
Homework Statement
I am working through chapter 47 of the Landau Lifschitz. And there is the following argument:
The vector potential is a function of t - \frac{x}{c}
From the defining equations for the electric and magnetic fields:
\vec{E} = - \frac{1}{c} \frac{\partial \vec{A}}{\partial t}, \vec{B} = \nabla \times \vec{A}
follows
\vec{E} = - \frac{1}{c} \vec{A'}
\vec{B} = \nabla \times \vec{A} = \nabla (t- \frac{x}{c}) \times \vec{A'} = - \frac{1}{c} \vec{n} \times \vec{A'}
\vec{B} = \vec{n} \times \vec{E}
I can't follow his argument.
Why did the equation for the electric field change from a time derivative of A to a derivative over (t- x/c).
Let's define t_r\equiv t-\frac{x}{c}. Then apply the chain rule:
\begin{aligned}\frac{\partial}{\partial t}\textbf{A}(\textbf{x},t_r) &= \frac{\partial t_r}{\partial t}\frac{\partial}{\partial t_r}\textbf{A}(\textbf{x},t_r)+ \frac{\partial \textbf{x}}{\partial t}\cdot\mathbf{\nabla}\left(\textbf{A}(\textbf{x},t_r)\right) \\ &= (1)\textbf{A}'(\textbf{x},t_r)+(\mathbf{0})\cdot\mathbf{\nabla}\left(\textbf{A}(\textbf{x},t_r)\right) \\ &= \textbf{A}'(\textbf{x},t_r)\end{aligned}
And where does that \nabla (t - x/c) come from?Finally where does that vector n come from?
Again, use the chain rule: In index notation w/ Einstein summation convention,
\begin{aligned}\mathbf{\nabla}\times\textbf{A}(\textbf{x},t_r) &= \textbf{e}_i\epsilon_{ijk}\partial_j A_k(\textbf{x},t_r) \\ &= \textbf{e}_i\epsilon_{ijk}\left[(\partial_j\textbf{x})\cdot\mathbf{\nabla}A_k(\textbf{x},t)\right]_{t=t_r} + \textbf{e}_i\epsilon_{ijk}\frac{\partial}{\partial t_r} A_k(x_m\textbf{e}_m,t_r) \partial_j t_r \\ &= \textbf{e}_i\epsilon_{ijk}\left[\textbf{e}_j\cdot\mathbf{\nabla}A_k(\textbf{x},t)\right]_{t=t_r}+ (\mathbf{\nabla}t_r)\times\textbf{A}(x_m\textbf{e}_m,t_r) \\ & = \left[\mathbf{\nabla}\times\textbf{A}(x_m\textbf{e}_m,t)\right]_{t=t_r} -\frac{1}{c} (\mathbf{\nabla}x)\times\textbf{A}(x_m\textbf{e}_m,t_r) \\ & =\end{aligned}
I don't have the text with me, but it looks like Landau is describing a case where radiation is emitted in the \textbf{n}\equiv \mathbf{\nabla}x direction, and you are only interested in a region where the non-retarded magnetic field is zero.
