A semi-circular wire of radius $R$ is connected to a wire bent in the form of a sine curve to form a

Question

A semi-circular wire of radius $R$ is connected to a wire bent in the form of a sine curve to form a closed loop as shown in the figure. If the loop carries a current $I$ and is placed in an uniform magnetic field $B$, then the total force acting on the sine curve is

Professor PC Thomas Classes · Answer

Since the magnetic force on a wire in a uniform field depends only on the straight-line displacement vector between its endpoints (a length of $2 R$ from $S$ to $P$ ), the force on the sine curve evaluates to the equivalent straight-wire force $F=I(2 R) B=2 B I R$ directed upward $(\operatorname{via} F=I L 	imes B)$.