# Finding induced current

1. Mar 25, 2006

### xman

i am trying to solve for the induced current for the following situation: a magnetic dipole, $$\vec{m}=m_{0}\hat{z}$$ moving with constant velocity coaxially with thin conducting ring. the ring has a resistance r and self-inductance L.

what i've done is take the magnetic field of the dipole
$$\vec{B}=\frac{\mu_{0}m_{0}}{4\pi r^{3}} \left(2\cos(\theta)\hat{r}+\sin(\theta)\hat{\theta}\right)$$
and imaging the spherical cap of radius $$R$$ which is the same radius as the conducting ring, and dotting this area element with the magnetic field of the dipole, since the divergence of the magnetic field is zero, this should work. then if i integrate i get the flux and emf i have
$$\phi_{m} = \frac{\mu_{0}m}{2}\frac{R^{2}}{\left(R^{2}+z^{2}\right)^{3/2}} \Rightarrow \varepsilon =-\dot{\phi}_{m}=\frac{3}{2}\mu_{0}m R^{2}\frac{z \dot{z}}{\left(R^{2}+z^{2}\right)^{5/2}}$$
now to find the current i have the ode
$$I\,r= -L\dot{I}+\frac{3}{2}\mu_{0}m R^{2}\frac{z \dot{z}}{\left(R^{2}+z^{2}\right)^{5/2}}$$
which i have been unable to solve, which of course is my question. Any suggestion is always greatly appreciated thanks.

Last edited: Mar 25, 2006