Question

A metal ring is oriented with the plane of its area perpendicular to a spatially uniform magnetic field that increases at a steady rate. If the radius of the ring is doubled, by what factor do (a) the emf induced in the ring and (b) the electric field induced in the ring change?

Short answer

1. The emf induced in the ring increases by a factor of four when the radius of the ring is doubled.
2. The electric field is induced twice in the ring as the radius of the ring is doubled.

Step-by-step solution

Step 1:

1. The induced emf is given by the Faraday’s law;

Hence, the induced emf grows fourfold when the radius is doubled since it is proportional to the square of the radius.

Step 2:

1. The induced electric field can be estimated by calculating the work done in transporting a test charge around the loop. It exerts an electrostatic force $\mathrm{F}=\mathrm{qE}$and performs work $\mathrm{F}\times 2\mathrm{\pi r}$by moving a test around the loop. However, the completed work can alternatively be written as.
2. $\mathrm{E}=-\frac{\mathrm{r}}{2}\frac{\mathrm{dB}}{\mathrm{dt}}$

Hence, the induced electric field is proportional to the ring's radius. When the radius is twice, so does the electric field.