- #1
dEdt
- 288
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I'm having some trouble confirming Ampere's law for a moving point charge.
Let's say we have a point charge [itex]q[/itex] moving with velocity [itex]\mathbf{v}[/itex]. The magnetic field it creates is given by
[tex]\mathbf{B}=\frac{\mu_0 q}{4\pi r^3} \mathbf{v}\times \mathbf {r}.[/tex]
Now consider a circular loop centred on the point charge and perpendicular to its velocity. Then
[tex]\oint \mathbf{B}\cdot d \mathbf{r}=\frac{\mu_0 q v}{2r}.[/tex]
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?
Let's say we have a point charge [itex]q[/itex] moving with velocity [itex]\mathbf{v}[/itex]. The magnetic field it creates is given by
[tex]\mathbf{B}=\frac{\mu_0 q}{4\pi r^3} \mathbf{v}\times \mathbf {r}.[/tex]
Now consider a circular loop centred on the point charge and perpendicular to its velocity. Then
[tex]\oint \mathbf{B}\cdot d \mathbf{r}=\frac{\mu_0 q v}{2r}.[/tex]
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?