1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Ampere's law for a point charge

  1. Jun 2, 2014 #1
    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?
  2. jcsd
  3. Jun 2, 2014 #2


    User Avatar

    Staff: Mentor

    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:
    $$\oint {\vec B \cdot d \vec l} =
    \mu_0 \int {\vec J \cdot d \vec a} +
    \mu_0 \epsilon_0 \frac{d}{dt} \int {\vec E \cdot d \vec a}$$
  4. Jun 2, 2014 #3
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook