A very long, straight solenoid with a cross-sectional area of 2.24 cm2 is wound

Question

A very long, straight solenoid with a cross-sectional area of 2.24 cm2 is wound with 90.7 turns of wire per centimeter. Starting at t = 0, the current in the solenoid is increasing according to i(t)= ( 0.164 A/s2 )t2. A secondary winding of 5 turns encircles the solenoid at its center, such that the secondary winding has the same cross-sectional area as the solenoid.

What is the magnitude of the emf induced in the secondary winding at the instant that the current in the solenoid is 3.2 A ?

Express your answer with the appropriate units.

Explanation / Answer

at I = 3.2

3.2 = 0.164*t^2

t = 4.42 s

B = uo*n*I

n = number of turns /m = 90.7*100 = 9070 turns/m

I = 0.164t^2

for secondary coil

number of turns N2 = 5

A2 = A1 = 2.24*10^-4 m^2

flux = N2*B*A2 = N2*uo*n*I*A2

emf = rate of flux

e = N2*uo*n*A2*(dI/dt)

e = 5*4*pi*10^-7*9070*2.24*10^-4*0.164*2t

e = 5*4*pi*10^-7*9070*2.24*10^-4*0.164*2*4.42

e = 1.85*10^-5 V

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