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A circular loop of radius r spins about the z-axis with angular velocity omega; its normal stays perpendicular to z (in the x-y plane). At t = 0 the normal points along +y. A constant external field B = B_y j + B_z k is present. What is the induced EMF, epsilon(t), in the loop?
- pi r² omega B_y sin(omega t)
- pi r² omega B_z cos(omega t)
- pi r² omega B_z sin(omega t)
- pi r² omega B_y cos(omega t)
Correct answer: pi r² omega B_y sin(omega t)
Solution
The normal rotates in the x-y plane: n = cos(omega t) j - sin(omega t) i (starting along +y). Flux Phi = (pi r²)(B. n) = pi r² B_y cos(omega t) (B_z k contributes nothing since n has no z-component). EMF = -dPhi/dt = pi r² omega B_y sin(omega t).
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