Question

A long solenoid (n = 1500 turns/m) has a cross-sectional area of 0.40 m² and a current given by I = 3.0t A, where t is in seconds. A flat circular coil (N = 500 turns) with a cross-sectional area of 0.15 m² is inside and coaxial with the solenoid. What is the magnitude of the emf induced in the coil at t = 2.0 s?

Answer 1

Given informations in the problem :

Number of turns of the bigger solenoid (n) = 1500 turns/m

Number of turns of the flat coil (N) = 500 turns/m

Cross-sectional area of bigger solenoid (A) =  0.40 m²

Cross-sectional area of 0.15 m²

Current : 3.0t A, where t is in seconds

We know from the relation of coaxial solenoid,

M = μ₀ n N A

Where μ₀ = 4 * 10 ^-7 H m^{- 1}

M is the mutual induction in Henry

M = 4 * 10 ^-7 * 1500 * 500 * 0.4

= 0.3768 Henry

Now we need to find the magnitude of the emf induced in the coil at t = 2.0 s

For this we will use the formula

E = -M dI/dt

where ,

M is the mutual induction

and dI/dt is the rate of change of current with time

so change in current can be calculated as follows:

Initially when t = 0 , then I = 0 Amp

After 2 seconds, t = 2 then I = 3*2 = 6 Amp

therefore , dI/dt is 6-0/2-0 = 3 A/sec

now,

Induced EMF (E) = 0.376 * 3 = 1.128 Volts.

Therefore, the magnitude of the emf induced in the coil at t = 2.0 s is 1.128 Volts.