Consider a magnetic field with field lines perpendicular to a loop of 15cm in ra

Question

Consider a magnetic field with field lines perpendicular to a loop of 15cm in radius. The magnetic field increases linearly with time going from zero to 0.8T in 20secs a. Provide an expression for the magnetic flux through this surface as a function of time b. Calculate the electromotive force induced in the loop c. Assuming the total resistance of the loop is 5 mu ohm, what is the current induced in the loop? d. Looking at the loop in the same direction of the magnetic field, is the induced current clockwise or counterclockwise? What if the magnetic flux were to decrease with time? e. Calculate the total magnetic field at t = 20 secs (remember that the total magnetic field is equal to the sum of the external field plus the magnetic field produced by the induced current in the loop) f. Does the total magnetic field increases or decreases if the resistance of the loop decrease? Explain why g. For what value of the loop resistance the total magnetic field is zero? h. Assuming that the magnetic field is left constant after 20secs, what is the electromotive force induced in the loop for t > 20secs?

Explanation / Answer

a)phi(t) = A B(t)

b)we know that induced emf is given by

e = -d(phi)/dt

e = -d(BA)/dt

e = -A dB/dt = A (B2 - B1)/t

e = 3.14 x 0.15^2 (0.8 - 0)/20 = 0.00283 V

Hence, e = 0.00283 V = 2.83 mV

c)R = 5 x 10^-6 Ohm

from Ohm's law, V = IR => I = V/R = e/R

I = 0.00283/5 x 10^-6 = 565.2 A

Hence, I = 565.2 A

d)B(t = 20) = 0.8 A

B' = u0 I/2 R

B' = 4 pi x 10^-7 x 565.2/ 2 x 0.15 = 0.004 T

B(tot) = 0.8 - 0.0024 = 0.798 T

f)with the increase in resistance the current decreases and henve the B field decreases.

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