Ampere's law for a point charge

[dEdt]

I'm having some trouble confirming Ampere's law for a moving point charge.

Let's say we have a point charge $q$ moving with velocity $\mathbf{v}$. The magnetic field it creates is given by
$$\mathbf{B}=\frac{\mu_0 q}{4\pi r^3} \mathbf{v}\times \mathbf {r}.$$

Now consider a circular loop centred on the point charge and perpendicular to its velocity. Then
$$\oint \mathbf{B}\cdot d \mathbf{r}=\frac{\mu_0 q v}{2r}.$$

By Ampere's law, this is proportional to the rate that charge passes through the surface of the closed loop. But this latter quantity is a Dirac delta function, so it seems that Ampere's law doesn't work for point charges!? What did I do wrong?

 

[jtbell]

A moving point charge produces not just a magnetic field, but also an electric field which varies with time at any point. Therefore you have to use Maxwell's generalization of Ampere's Law that includes the rate of change of electric flux through the loop:
$$\oint {\vec B \cdot d \vec l} =
\mu_0 \int {\vec J \cdot d \vec a} +
\mu_0 \epsilon_0 \frac{d}{dt} \int {\vec E \cdot d \vec a}$$

 

[dEdt]

Thanks.