Find the torque and the force between two circular loops of wire, carrying the same currents I, and of the same radius R, when they are located a distance L apart, with L >> R, and with their axes parallel and the currents in the same direction. Express the torque and the force in terms of the angle θ between their axes and their line of centers.
Text Solution
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Sol. With L >> R, the two loops act like two magnetic dipoles with magnetic moment µ 0 IA, where A is the area of the loop. The induction B at loop two due to loop one can be resolved into two components; B r , in the direction of increasing r, and B θ in the direction of increasing θ . We find
B r = 
IR 2 
= 
IR 2 
B θ = 
IR 2 
= 
IR 2 
where x = L cos θ , y = L sin θ , and µ 0 I π R 2 is the equivalent dipole moment of each loop. The torque on loop two is
= I 
× 
( = µ 0 I 
× H)
or τ = I π R 2 (B r sin θ + B θ cos θ )
= 
The direction of the torque is pointing into the plane of the paper. The force on loop two is
= 
+ 
where F θ = IR 2 
= – 
I 2 R 2 y 
= – μ 0 (IR 2 ) 2 
= – μ 0 (IR 2 ) 2 
and F r = IR 2 
= 
μ 0 (IR 2 ) 2 
= 
μ 0 (IR 2 ) 2 
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