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Next: Gauge transformations Up: Time-dependent Maxwell's equations Previous: Faraday's law


Electric scalar potential?

Now we have a problem. We can only write the electric field in terms of a scalar potential (i.e., ${\bf E} = -
\nabla\phi$) provided that $\nabla\times{\bf E} = {\bf0}$. However, we have just found that in the presence of a changing magnetic field the curl of the electric field is non-zero. In other words, ${\bf E}$ is not, in general, a conservative field. Does this mean that we have to abandon the concept of electric scalar potential? Fortunately, no. It is still possible to define a scalar potential which is physically meaningful.

Let us start from the equation

\begin{displaymath}
\nabla \cdot {\bf B} = 0,
\end{displaymath} (379)

which is valid for both time-varying and non time-varying magnetic fields. Since the magnetic field is solenoidal, we can write it as the curl of a vector potential:
\begin{displaymath}
{\bf B} = \nabla\times{\bf A}.
\end{displaymath} (380)

So, there is no problem with the vector potential in the presence of time-varying fields. Let us substitute Eq. (380) into the field equation (376). We obtain
\begin{displaymath}
\nabla\times{\bf E} = - \frac{\partial  \nabla\times{\bf A}}{\partial t},
\end{displaymath} (381)

which can be written
\begin{displaymath}
\nabla\times\left( {\bf E} + \frac{\partial {\bf A} }{\partial t} \right) ={\bf0}.
\end{displaymath} (382)

We know that a curl-free vector field can always be expressed as the gradient of a scalar potential, so let us write
\begin{displaymath}
{\bf E} + \frac{\partial {\bf A} }{\partial t} = -\nabla\phi,
\end{displaymath} (383)

or
\begin{displaymath}
{\bf E} = - \nabla\phi - \frac{\partial {\bf A} }{\partial t}.
\end{displaymath} (384)

This is a very nice equation! It tells us that the scalar potential $\phi$ only describes the conservative electric field generated by electric charges. The electric field induced by time-varying magnetic fields is non-conservative, and is described by the magnetic vector potential ${\bf A}$.


next up previous
Next: Gauge transformations Up: Time-dependent Maxwell's equations Previous: Faraday's law
Richard Fitzpatrick 2006-02-02