A very long, but very thin solenoid (radius, $a$) is placed at the center of a large ring with resistance $R$ and radius $b$. Note here that $b»a$. A current runs through the solenoid that changes with time, $I(t)$.

As you have seen, a current will develop in the ring. Show that the energy is carried from the solenoid to the ring through the Poynting vector, $\mathbf{S}$.

The power dissipated in the ring should be equal to the power delivered by the Poynting vector ($\iint \mathbf{S} \cdot d\mathbf{A}$). Show by integrating over the whole solenoid that this is the case, you will need to use the magnetic field produced by each ring of the solenoid.