A circular loop of wire of radius 0.5 m and resistance 0.015 lies in the plane o

Question

A circular loop of wire of radius 0.5 m and resistance 0.015 lies in the plane of the page. A magnetic field directed out of the page increases in magnitude linearly with time from 0.0 T at t = 0 ms to 1.2 T at t = 6 ms. What is the magnitude of the electromotive force that is induced in the loop? What is the magnitude of the electric current that is induced in the loop? What is the magnitude of the magnetic field that the current in the loop creates at the center of the loop? At t = 3 ms, what is the magnitude of the total magnetic field at the center of the loop?

Explanation / Answer

= - N d/dt = - d/dt since N, the number of loops is 1.

= - d/dt = - d( BACos) /dt

where B is the magnetic field, A is the area of the loop and is the angle between the normal of the area and the magnetic field.

= - d (BA) /dt since = 0

now, the magnetic field changed fron B = 0 T to B= 1.2 T in a time of dt = 6ms

= - d (BA) /dt = - (1.2 A - 0) / (6 x 10-3 ) = - ( 1.2 x 0.52 ) / 0.006 = - 157.0796327 volts

the negative sign just to get the direction. it indicates that the induced emf always tries to oppose the change in the magnetic flux.

So, the magnitude of the induced emf is 157.0796327 volts

magnitude of the current induced = induced emf / resistance = 157.0796327 / 0.015 = 10471.97551 Ampere

Magnetic field at the center of a current loop is given by B = oI/2r      

so the magnetic field B' due to the induced current will be B' = 4 × 107 x 10471.97551 / 2 x 0.5

B' = 0.01315947253 T

Since the induced current always tries to oppose the change, the direction of this B' will be directed into the page.

At, t = 3 ms, the magnetic field directed out of the page will be 0.6 T (Since the magneitc field increases linearly from 0 to 1.2 T in a time of t= 0 to t = 6ms)

So, at t = 3ms, the net magnetic field at the center will be Bnet = B - B' = 0.6 - 0.01315947253

= 0.5868405275 T

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