Answer to Question #68816 in Electricity and Magnetism for Nathaniel

Question

A 30 turn circular coil of radius 4cm and resistance 1 ohms is placed in a magnetic field directed perpendicularly to the plane of the coil. the magnitude of the magnetic field varies in time according to the expression B = 0.01t + 0.04t^2 T. calculate the induced electro magnetic field in the coil at t = 5 seconds

Answer 1

Let’s use the Faraday’s law and find the induced emf in the coil:
E=-(dΦ_B)/dt,
here, E is the emf generated between the ends of the coil, Φ_B=NBAcosθ=NBAcos0^°=NBA is the magnetic flux through the coil, N is the number of turns of the coil, B is the magnetic field, A=πr^2 is the cross-sectional area of the coil, r is the radius of the circular coil, θ is the angle between the magnetic field and the normal to the plane of the coil (since the magnetic field directed perpendicularly to the plane of the coil, θ=0^°).
Then, we get:
E=-(dΦ_B)/dt=-d(NBA)/dt=-NA dB/dt=-Nπr^2 d/dt (0.01t+0.04t^2 )==-Nπr^2 (0.01+0.08t),
E(t=5 s)=-30∙π(0.04 m)^2 (0.01+0.08∙5 s)=-0.0618 V=-61.8 mV.
Answer:
E(t=5 s)=-61.8 mV.