The diagram at right shows a square loop of wire, being pulled at constant speed

Question

The diagram at right shows a square loop of wire, being pulled at constant speed in the direction shown; the dimensions of the square, and the speed, are indicated. The total resistance of the loop is 0.1 ohm. There is a uniform magnetic field of magnitude 0.8 T, pointing into the page, to the right of the dotted line; to the left of the dotted line the magnetic field is zero. At time t = 0 the right corner of the loop first enters the magnetic field. (a) Determine the induced current in the loop, as a function of time. (b) Determine the maximum value of the induced current in the loop, and also determine where the loop is when the current is a maximum (specify it by how far into the magnetic field the right corner of the loop is at that instant).

Explanation / Answer

here ,

for the loop

length of diagonal = sqrt(2) * 10 = 14.14 cm

Now, untill the half diagonal is inside

t = .1414/(2 * 10) = .00707 s

for 0 < t < 7.07 *10^-3 s

perpendicular length entering the magnetic field

L = (v * t) * tan(45) * 2

L = 2 * v * t

induced current in the loop = 2 * vt * v * B/R

induced current in the loop = 2 * 10^2 * 0.80 * t/0.10

induced current in the loop = 1600 * t

Now, for 7.07 *10^-3 < t < 2 * 7.07 *10^-3

perpendicular length , L = 2 *(0.1414 - v * t)

induced current in the loop = 2 *(0.1414 - v * t) * v * B/R

induced current in the loop = 2 * (0.1414 - 10 * t) * 0.80* 10/0.10

induced current in the loop = 160 * (0.1414 - 10 * t)

after that for t > 2 * 7.07 *10^-3 s

induced current = 0 A

b)

maximum induced current in the loop = 0.1414 * 10 * 0.80/.10

maximum induced current in the loop = 11.3 A

the current is maximum when half the loop is inside , when upper and lower corners are entering the magnetic field

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