Question

Deduce the expression for the torque `vec"τ"` acting on a planar loop of area `vec"A"` acting on a planar loop of area `vec"B"`. If the loop is free to rotate, what would be its orientation in stable equilibrium?

Answer 1

(a) The area vector of the loop ABCD makes an arbitrary angle θ with the magnetic field.

(b) Top view of the loop. The forces F₁ and F₂ acting on the arms AB and CD are indicated.

We consider the case when the plane of the loop, is not along the magnetic field but makes an angle θ with it. Fig. (a) illustrates this general case. The forces on the arms BC and DA are equal, opposite, and act along the axis of the coil, which connects the centers of mass of BC and DA. Being collinear along the axis they cancel each other, resulting in no net force or torque. The forces on arms AB and CD are F₁and F₂. They too are equal and opposite, with magnitude,
F₁ = F₂ = I b B
But they are not collinear! This results in a couple of Fig. (b) is a view of the arrangement from the AD end and it illustrates these two forces constituting a couple. The magnitude of the torque on the loop is,

`"τ" = "F"_1 "a"/(2) sin θ  + "F"_2 "a"/(2) sin θ`

= I ab B sin θ

= I A B sin θ         ....(i)

As θ → 0, the perpendicular distance between the forces of the couple also approaches zero. This makes the forces collinear and the net force and torque zero. The torques in the above equation can be expressed as vector products of the magnetic moment of the coil and the magnetic field. We define the magnetic moment of the current loop as,
m = I A
where the direction of the area vector A is given by the right-hand thumb rule and is directed into the plane of the paper in Fig. (a). Then as the angle between m and B is `theta`, equation (i) can be expressed by one expression

`vec"τ" = vec"m" xx vec"B"`

we see that the torque `vec"τ"    "vanishes when"   vec"m"` is either parallel or antiparallel to the magnetic field  `vec"B"` This indicates a state of equilibrium as there is no torque on the coil (this also applies to any object with a magnetic moment `vec"m").`

When `vec"m" and vec"B"` re parallel the equilibrium is a stable one.