Chapter 29, Problem 29.58P

CALC A circular conducting ring with radius r₀ = 0.0420 m lies in the .xy-plane in a region of uniform magnetic field $\stackrel{\to }{B}=B_0\left[1-3{\left(t/t_0\right)}^2+2{\left(t/t_0\right)}^3\right]\hat{K}$. In this expression, t₀ = 0.0100 s and is constant, t is time, $\hat{K}$ is the unit vector in the +z-direction, and B₀ = 0.0800 T and is constant. At points a and b (Fig. P29.58) there is a small gap in the ring with wires leading to an external circuit of resistance R = 12.0 Ω There is no magnetic field at the location of the external circuit. (a) Derive an expression, as a function of time, for the total magnetic flux Ф_B through the ring. (b) Determine the emf induced in the ring at time t = 5.00 × 10⁻³s. What is the polarity of the emf? (c) Because of the internal resistance of the ring, the current through R at the time given in part (b) is only 3.00 mA. Determine the internal resistance of the ring. (d) Determine the emf in the ring at a time t = 1.21 × 10⁻² s. What is the polarity of the emf? (e) Determine the time at which the current through R reverses its direction.

Figure P29.58