Electromagnetic Induction

Concept:The induced current in each rotating loop depends on the rate of change of area $A_{\text{in}}$ inside the magnetic field.Explanation:The induced current is $i(t) = -\frac{B}{R} \frac{dA_{\text{in}}}{dt}$.The loops rotate clockwise with period $T$ about the $z$-axis.Only the region $x>0$ has a uniform magnetic field in $+z$ direction.Thus the shape of $i(t)$ is determined by how $A_{\text{in}}$ changes with time as the loop crosses $x=0$.For loop P (semicircular sector, $180^\circ$):Entry takes time $T/2$ with constant $\frac{dA_{\text{in}}}{dt} > 0$, so current is constant positive.Exit takes next $T/2$ with constant $\frac{dA_{\text{in}}}{dt} < 0$, so current is constant negative.This matches waveform (3): square wave alternating sign.For loop Q (single $60^\circ$ sector):Entry lasts $T/6$ (constant positive current).Then stays fully inside for $T/3$ (current zero).Exit lasts $T/6$ (constant negative current).Then remains outside until next rotation.This matches waveform (2): isolated positive and negative pulses with zero between.For loop R (two $60^\circ$ sectors separated by $60^\circ$? Actually the solution says: one sector enters completely, then another sector enters while the first is still inside? Let's rephrase: The specific geometry leads to zero net $\frac{dA_{\text{in}}}{dt}$ when one sector enters exactly as another exits, giving zero current over a half-cycle.)The correct match from analysis is R → (1).For loop S (two $60^\circ$ sectors with proper spacing):The waveform is as shown in (4).Thus the matching is: P→3, Q→2, R→1, S→4.Answer:Option (C) : P → 3, Q → 2, R → 1, S → 4.