Torque on a current carrying rectangular loop in a magnetic field|Magnetism

Torque on a current carrying rectangular loop in a magnetic field

Consider a rectangular loop ABCD being suspended in a uniform magnetic field B and direction of B is parallel to the plane of the coil as shown below in the figure

Magnitude of force on side AM according to the equation(13) is
F_{AB}=IhB ( angle between I and B is 90^{0})
And direction of force as calculated from the right hand palm rule would be normal to the paper in the upwards direction

Similarly magnitude of force on CD is
F_{CD}=ihB
and direction of F_{CD} is normal to the page but in the downwards direction going into the page

The forces F_{AB} and F_{CD} are equal in magnitude and opposite in direction and hence they constitute a couple

Torque τexerted by this couple on rectangular loop is
τ=IhlB
Since torque = one of the force * perpendicular distance between them

No force acts on the side BC since current element makes an angle θ=0 with B due to which the product (ILXB) becomes equal to zero

Similarly on the side DA ,no magnetic force acts since current element makes an angle θ=180^{0} with B

Thus total torque on rectangular current loop is
τ=IhlB
=IAB --(15)
Where A=hl is the area of the loop

If the coil having N rectangular loop is placed in magnetic field then torque is given by
τ=NIAB ----(16)

Again if the normal to the plane of coil makes an angle θ with the uniform magnetic field as shown below in the figure then

τ=NIABsinθ

We know that when an electric dipole is placed in external electric field then torque experienced by the dipole is
τ=P X E=PEsinθ
Where P is the electric dipole moment

comparing expression for torque experienced by electric dipole with the expression for torque on a current loop i.e ,
τ=(NIA)Bsinθ
if we take NIA as magnetic dipole moment (m) analogous to electric dipole moment (p),we have m=NIA -- (18)
then
τ=m X B -- (19)

The coil thus behaves as a magnetic dipole

The direction of magnetic dipole moment lies along the axis of the loop

This torque tends to rotate the coil about its own axis .Its value changes with angle between the plane of the coil and the direction of the magnetic field

Unit of magnetic moment is Ampere.meter^{2} (Am^{2})

Equation (18) and (19) are obtained by considering a rectangular loop but these equations are valid for plane loops of any shape