A uniform magnetic field is applied perpendicular to the plane of a 60-turn circular coil with a radius of 6.0 cm and a resistance of 0.60 Ω. If the magnetic field increases uniformly from 0.20 T to 1.80 T in 0.20 s, what is the magnitude of the emf induced in the coil?

The expression for the magnetic flux for a given area is given by,

ϕ=BAcosθ

The surface is perpendicular to the field then the angle is zero.

The expression for the magnetic flux is given by,

ϕ=BA

For calculating the change in magnetic flux, differentiate the above equation with respect to time.

The expression for the change in magnetic flux is given by,

..... (1)

Here, A is area and it is constant.

Area of the circular coil is given by πr²  

The expression for the change in magnetic flux in terms of the radius of the circular coil is given by,

....... (2)

Here, r is the radius of the coil.

The magnetic field increases uniformly from 0.20 T to 1.8 T in 0.20 s.

The magnetic field increases uniformly from 0.20 T to 1.8 T in 0.20 s.

Substitute 6.0 cm for r (1.8T−0.20T) for dB and 0.20 s for dt  in the equation (2).

Explanation:

Magnetic flux is directly proportional to the magnetic field and the area of the flux through it. The area remains the same as the magnetic flux is increased by increasing the magnetic field.

The expression for the EMF is given by,

ε=−Ndϕ /dt ...... (3)

The value of change in magnetic flux with respect to time is calculated in step 1. Thus, we are using the value of = 0.090432T/s in the above equation (3)

Substitute 0.090432T/s  for dt /dϕ  and 60 for N in the equation (3).

ε=−(60)(0.090432T/s)  

=5.425 V

The magnitude of induced EMF in the coil is,

∣ε∣=5.425 V

The magnitude of the EMF induced in the coil is 5.425 V.

Explanation:

Lenz’s law is defined as

ε ∝  dt /dϕ

To remove the proportionality sign, a constant of proportionality is introduced.

ε=−N dt /dϕ

Here, N is the number of turns in the coil.

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