A circular loop of radius and negligible resistance

PHYSICS · Subjective · AP · Physics 2: Algebra-based · Magnetism and Electromagnetism · Magnetism and Electromagnetism

Problem

A circular loop of radius rr and negligible resistance, carrying NN tightly wound turns, is set into rotation about a diameter in a uniform magnetic field BB according to the angular displacement law θ=12αt2\theta = \frac{1}{2}\alpha t^{2}, where α\alpha is a constant angular acceleration and θ\theta is measured from the position in which the plane of the loop is perpendicular to the magnetic field. (a) Express the magnetic flux linked with the loop as a function of time. (b) Derive an expression for the instantaneous induced emf ε(t)\varepsilon(t) in the loop. (c) At t=t1t = t_{1}, show that the magnitude of the induced emf is ε(t1)=NBπr2αt1sin⁡(12αt12)\varepsilon(t_{1}) = N B \pi r^{2} \alpha t_{1} \sin \left( \frac{1}{2} \alpha t_{1}^{2} \right). (d) For N=100N = 100, r=5.0 cmr = 5.0\,\text{cm}, B=0.15 TB = 0.15\,\text{T} and α=10 rad s−2\alpha = 10\,\text{rad\,s}^{-2}, compute the value of ε\varepsilon at t=0.20 st = 0.20\,\text{s} and state whether the induced current (if the loop is part of a closed circuit) will oppose or assist the increase in θ\theta.

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