A circular metal ring, as shown on the diagram below, is constructed so as to ex

Question

A circular metal ring, as shown on the diagram below, is constructed so as to expand or contract freely. In a region with a constant magnetic field Bo oriented perpendicular to it, the ring expands, with its radius growing with time as r=ro(1+at2). As the ring expands and grows thinner, its resistance per unit length changes according to R=Ro(1+ßt2). Find the current induced in the ring as a function of time. To check your answer, suppose that Bo = 6.70 mT, ro = 15.0 cm, Ro = 1.85 [ ?/(m)], a = 0.115 × 10-4 s-2, and ß = 0.495 × 10-2 s-2. What is the value of the induced current at t = 85.0 s? (Note: Give the direction of the current where when viewed from above a positive current will move counterclockwise.)

Explanation / Answer

Bo = 6.70 mT, ro = 15.0 cm, Ro = 1.85 Ohm/m, a = 0.115 × 10^-4 s^-2, and ß = 0.495 × 10^-2 s^-2. What is the value I of the induced current at t = 85.0 s? resistance RR = R*2*pi*r area A = pi*r^2 magnetic flux F = Bo*A = Bo*pi*r^2 emf E = -dF/dt = -Bo*pi*2*r*dr/dt = -Bo*pi*2r*ro*2at current I = E/RR = -Bo*pi*2r*ro*2at/(R*2*pi*r) = -Bo*ro*2at/R = -Bo*ro*2at/[Ro(1 + ßt^2)] = -2.89*10^-8 A

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