Torque on a Current Carrying Loop

The fact that a force exists on a wire carrying a current in a magnetic field has an interesting application when we consider a current carrying loop. Consider a rectangular loop carrying a current $\vec{I}$ in a magnetic field $\vec{B}$ as in Fig. 1.5.

  

Figure 1.5: Torque on a current loop in a magnetic field

We consider the force $\vec{F}$ = l$\vec{I}$ x $\vec{B}$ of Eq.(1.4) on each of the four sides of the loop. On the top and bottom sections this force vanishes, because $\vec{I}$ and $\vec{B}$ are parallel or antiparallel in these cases. The force $\vec{F}_{L}^{}$ on the left section will have a magnitude aIB in the direction indicated, while the force $\vec{F}_{R}^{}$ on the right section will have the same magnitude aIB but in the opposite direction. These two forces, since they are oppositely directed, do not give rise to a net linear acceleration, but they do tend to rotate the loop around the vertical axis. There will thus be a net torque $\tau$ on the loop, which is given by

$$\tau {\frac{b}{2}} {\frac{b}{2}}$$ (5)

where A = ab is the area of the loop.

Although we have considered a rectangular loop, the relation $\tau$ = BIA holds for an arbitrarily shaped loop of area A . The fact that current carrying loops experience a net torque in a magnetic field is the principle behind the electric motor, where the electrical energy involved in establishing a current is converted into the mechanical energy of rotating a shaft.