An elastic conducting material is stretched into a circular loop of radius r0 =

Question

An elastic conducting material is stretched into a circular loop of radius r0 = 10 cm. It is placed with its plane perpendicular to a uniform magnetic eld of B = 0.90 T that acts into the page. When released, the radius of the loop starts to shrink at a constant rate of 2 mm/s

a. Write an expression for the radius, r, as a function of time, t.

b. Write an expression for the magnetic ux, B, as a function of time, t.

c. Write an expression for the induced emf, , in the loop as a function of t. Determine at t = 10 s.

d. If the resistance per unit length of the loop is 5 /m, what is value of the induced current in the loop
at t = 10 s? In what direction (clockwise or counterclockwise) does the current ow?

Explanation / Answer

r0 = 10 cm = 0.10 m

B = 0.90 T

dr/dt = 2 mm/s

dr/dt = 2 *10^-3 m/s

a) for the radius as a function of time

r = r0 + dr/dt * t

r = 0.10 + 2 *10^-3 * t

b)

magnetic flux = pi * r^2 * magnetic field

magnetic flux = pi * (0.10 + 2 *10^-3 * t )^2 * 0.90

c)

induced emf = d(magnetic flux)/dt

induced emf = d/dt(pi * (0.10 + 2 *10^-3 * t )^2 * 0.90)

induced emf = pi * (2 *10^-3) * (0.10 + 2 *10^-3 * t ) * 2* 0.90

induced emf = pi * (2 *10^-3) * (0.10 + 2 *10^-3 * 10 ) * 2* 0.90

induced emf = 1.357 *10^-3 V

d) at t = 10 s

induced current = induced emf/(resistance )

induced current = 1.357 *10^-3/(2 pi * (0.10 + 2 *10^-3 * 10 ) * 5 )

induced current = 3.6 *10^-4 A

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