Figure shows a circular wire-loop of radius a, carrying a current i, placed in a perpendicular magnetic field B.
(a) Consider a small part dl of the wire. Find the force on this part of the wire exerted by the magnetic field.
(b) Find the force of compression in the wire.
Given-
Radius of the circular wire = a
Electric current passing through the loop = i
Magnetic field Perpendicular to the plane = B
(a) Magnetic force due to presence of current on a small differential length dl given by –
where,
B= magnetic field
I = current
dl =differential length of the wire
and θ = the angle between B and dl
The direction of magnetic force,using Fleming’s left-hand rule is towards the centre for any differential length dl of the wire.
Also, dl and B are perpendicular to each other
(b) Suppose a part of loop subtends a small angle 2θ at the centre of a circular loop as shown in fig.
Then, looking into the fig. we can say
We know length l of an arc –
where ,
r is radius of the circle and the angle subtended by the arc at the center
here, the arc is subtending an angle 2θ
Since is small, sinθ will become negligible