A circular coil of 20 turns and radius 5.0 cm is placed with its plane oriented at 90° to a uniform magnetic field of The field is now increased at a steady rate, reaching a value of after 4.0 seconds. What emf is induced in the coil?

A circular coil of 20 turns and radius 5.0cm is placed with its plane oriented at 90° to a magnetic field of 0.1T. The field is now increased at a steady rate, reaching a value of 0.5T after 4 seconds. What emf is induced in the coil?

Answer:

0.01571V

Explanation:

Faraday's law of induction states that the induced emf (E) in a coil in a uniform magnetic field is proportional to the time (t) rate of change of magnetic flux (Ф) in the coil. The magnitude of the emf can be expressed mathematically as;

E = N [ΔФ / Δt] -----------------------(i)

Where;

N is the proportionality constant which also represents the number of turns in the loop

ΔФ = Ф₂ - Ф₁ [Ф₂ and Ф₁ are the final and initial values of the magnetic flux]

Δt = change in time

But;

magnetic flux Ф is given by

Ф = BAcosθ

Where;

B = magnetic field

A = Area of the coil = πr² [r = radius of the coil]

θ = Angle between the plane of the coil with respect to the magnetic field.

This implies that;

Ф₁ = B₁ Acosθ -----------------(ii)

Ф₂ = B₂ Acosθ ------------------(iii)

From the question;

r = radius = 5.0cm = 0.05m

A = π (0.05)² [Take π = 3.142]

A = 3.142 x (0.0025)

A = 0.007855m²

B₁ = initial value of the magnetic field = 0.1T

B₂ = final value of the magnetic field = 0.5T

θ = 0° [since the plane makes 90 degrees to the magnetic field, then it is parallel to the magnetic flux]

Therefore, substituting these values into equations (ii) and (iii) gives;

Ф₁ = 0.1 x 0.007855 x cos 0° = 0.0007855 Tm²

Ф₂ = 0.5 x 0.007855 x cos 0° = 0.0039275 Tm²

=> ΔФ = 0.0039275 - 0.0007855 = 0.003142

Also;

Δt = 4.0 seconds

N = 20

Substitute the values of ΔФ, Δt and N into equation (i) as follows;

E = 20 [0.003142 / 4]

E = 0.01571V

Therefore, the induced emf in the coil is 0.01571V