Question

A conducting single-turn circular loop with a total resistance of 7.50 Ω is placed in a time-varying magnetic field that produces a magnetic flux through the loop given by ΦB = a + bt2 − ct3, where a = 8.00 Wb, b = 15.5 Wb/s−2, and c = 7.50 Wb/s−3. ΦB is in webers, and t is in seconds. What is the maximum current induced in the loop during the time interval t = 0 to t = 1.55 s? (Enter the magnitude.)

Answer 1

Induced Emf is given by:

EMF = -N*d()/dt

= a + bt^2 - ct^3

d()/dt = 0 + 2*b*t - 3*c*t^2

d()/dt = 0 + 2b*t - 3c*t^2

Given that b = 15.5 & c = 7.50

Given that single-turn loop, So N = 1

Now induced Emf will be:

Emf = -1*(2b*t - 3c*t^2) = 3c*t^2 - 2b*t

Now induced current will be maximum when induced current is maximum, So Emf will be maximum when

d(Emf)/dt = d[3ct^2 - 2bt]/dt

d(Emf)/dt = 6ct - 2b = 0

t = b/3c = 15.5/(3*7.50) = 0.689 sec

Now at this time induced Emf will be:

Emf = 3*7.50*0.689^2 - 2*15.5*0.689 = -10.68 V

Max Emf = |-10.68 V| = 10.68 V

Now Using ohm's law induced current will be given by

Induced current = EMF/R

R = resistance of loop = 7.50 ohm

Using these values maximum induced current will be:

i_induced = 10.68/7.50

i_induced = 1.424 Amp

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