Electric Circuits
Beginner
Current in Magnetic Field
Source: High school physics (Chinese)
Problem
A square coil is wound from $N = 200$ turns of fine insulated wire, with each side of length $a = 150$ mm. It is placed in a uniform magnetic field of magnitude $B = 0.040$ T. When the coil carries a current of $I = 8.0$ A, find the maximum torque acting on the coil.
$\tau_{\max} = 1.44$ N$\cdot$m.
The torque on a planar current loop in a uniform field is
$$\tau = NIAB\sin\theta,$$where $A$ is the area of one turn and $\theta$ is the angle between $\vec{B}$ and the coil's normal. The torque is maximum when $\theta = 90^{\circ}$ (i.e., $\vec{B}$ lies in the plane of the coil):
$$\tau_{\max} = NIAB = NIa^2B.$$Substituting $N = 200$, $I = 8.0$ A, $a = 0.150$ m, $B = 0.040$ T:
$$\tau_{\max} = 200\times 8.0\times (0.150)^2 \times 0.040 = 200\times 8.0\times 0.0225 \times 0.040 = 1.44\ \text{N}\cdot\text{m}.$$