Force On A Curved Current-carrying Wire

The magnetic force on a current-carrying conductor in a uniform field is $\vec{F}=I\int d\vec{l}\times \vec{B}$. A powerful simplification applies for a uniform field: the net force on any curved wire equals the force on a straight wire joining its endpoints, because $\int d\vec{l}$ reduces to the straight-line vector $\vec{L}$ between the ends. For a semicircle of radius $R$, the two endpoints of the curved arc are separated by the diameter, so the effective length is $L=2R$. With the field perpendicular to the plane containing the wire, the angle between this effective length vector and $\vec{B}$ is $90^{\circ }$, giving $F=BI(2R)=2BIR$. The option $\pi BIR$ wrongly uses the full arc length $\pi R$, which would only matter if the field varied along the wire. The option $BIR$ uses $R$ instead of the diameter $2R$. The option $0$ would hold for a complete closed loop, where the endpoints coincide, but a semicircle is open. This effective-length theorem is a standard NCERT-level result. A check confirms that the force scales with the straight-line separation of the endpoints, here the diameter. This effective-length theorem dramatically simplifies otherwise tedious integrals and is well worth memorising for examination problems involving arcs, zigzags, and irregular bends in uniform fields. Its deeper meaning is that a uniform field exerts no net force on a closed current loop, since the closing endpoints coincide and the effective length vanishes; the loop instead experiences only a torque, which is exactly why current loops behave as magnetic dipoles that rotate rather than translate.