- #1

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The length gauge is obtained by the gauge transformation ##\mathbf{A} \rightarrow \mathbf{A} + \nabla \chi## with ##\chi = - \mathbf{r} \cdot \mathbf{A}##. Starting from the Coulomb gauge, we have

$$

\begin{align*}

\mathbf{A}_\mathrm{L} &= \mathbf{A}_\mathrm{C} + \nabla( -\mathbf{r} \cdot \mathbf{A}_\mathrm{C} ) \\

&= \mathbf{A}_\mathrm{C} - \mathbf{A}_\mathrm{C} = 0

\end{align*}

$$

How can I reconcile this with ##\mathbf{B} = \nabla \times \mathbf{A}##? It appears that there can be no magnetic field in the length gauge, while the gauge transformation leaves the magnetic field unchanged...