Magnetic Force on a Current-Carrying Wire — Question 5

Question 5

A semicircular arc and its closing diameter have equal and opposite net forces

A wire arc of radius R lies above the x axis and carries current I from (-R,0) to (R,0). A straight diameter returns the current from (R,0) to (-R,0). The field is uniform $B\hat z$.

1. Find the net force on the arc without assuming all its local force elements point in one direction.

2. Find the force on the diameter and the total force on the closed loop.

Question 5 - Solution

A semicircular arc and its closing diameter have equal and opposite net forces

(a) Method and calculation

$$\int_{\mathrm{arc}}d\vec\ell=2R\hat x.$$$$\vec F_{\mathrm{arc}}=I(2R\hat x)\times B\hat z=-2IRB\hat y.$$

(b) Interpretation and further calculation

$$\vec F_{\mathrm{diameter}}=I(-2R\hat x)\times B\hat z=+2IRB\hat y.$$$$\vec F_{\mathrm{total}}=0.$$

Physical check

The arc length is pi R, but the vector sum of its directed length elements is the diameter displacement.