A coil is wrapped with 206 turns of wire on the perimeter of a circular frame (of radius 31 cm). Each turn has the same area, equal to that of the frame. A uniform magnetic field is directed perpendicular to the plane of the coil. This field changes at a constant rate from 23 mT to 54 mT in 24 ms. What is the magnitude of the induced average E in the coil, over the time interval 24 ms during which the field changes? Answer in units of V.

  E = -80.33 V

Explanation:

For this exercise we use Faraday's law of induction

         E = - N d / dt

where the magnetic flux is

        \phi _{B} = B A cos θ

In the exercise they indicate that the magnetic field is perpendicular to the coil, therefore the normal of the coil area and the field has the same direction, consequently the angle is zero. Since the area is constant

        E = - N A dB / dt

let's look for each term

        a =π r²

they tell us that the variation in the field is linear

         dB / dT = (B_f-B₀) / t

we substitute

       E = - N (π r²) (B_f -B₀) / t

let's calculate

       E = - 206 π 0.31² (54 - 23) 10⁻³ / 24 10⁻³

        E = -80.33 V