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A wire is bent into three circular quadrants, each of radius r = 10 cm: quadrant ab lies in the xy-plane, bc lies in the yz-plane, and ca lies in the zx-plane, together forming a closed loop. A magnetic field B points in the positive x-direction and increases at the rate dB/dt = 3 mT/s. What is the magnitude of the emf induced in the wire?
- about 24 microvolt
- about 47 microvolt
- about 71 microvolt
- about 12 microvolt
Correct answer: about 24 microvolt
Solution
The flux is Phi = B. A_vec, where A_vec is the net vector area of the loop. Each quadrant contributes a vector area of magnitude (1/4)*pi*r² directed along its plane's normal. The quadrant in the yz-plane (bc) has its normal along x, so its area component along B is (1/4)*pi*r²; the quadrants in the xy and zx planes have normals along z and y and contribute zero x-component. Thus Phi = B*(1/4)*pi*r², and emf = (dB/dt)*(1/4)*pi*r².
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