A uniform magnetic field of `3000G` is established along the positive z-direction. A rectangular loop of sides `10cm` and `5cm` carries a current `12A`. What is the torque on the loop in the different cases shown in the figure. What is the force on each case? Which case corresponds to stable equilibrium?

Explanation

Magnetic field strength, `B = 3000G = 3000 xx 10^(-4)T = 0.3T`
Length of the rectangular loop, `l = 10cm`
Width of the rectangular loop, `b = 5cm`
Area of the loop,
` A= l xx b = 10 xx 5 = 50 cm^(2) = 50 xx 10^(-4)m^(2)`
Current in the loop, `I = 12A`
Now, taking the anti-clockwise direction of the current as positive and vise-versa:
(a) Torque, `vec(tau) = I vec(A) xx vec(B)`
From the given figure, it can be observed that A is normal to the y-z plane and B is directed along the z-axis.
`:. r = 12 xx (50 xx 10^(-4)) hati xx 0.3 hatk`
`=-0 1.8 xx 10^(-2)hatj Nm`
The torque is `1.8 xx 10^(-2)Nm` along the negative y-direction. The force on the loop is zero because the angle between A and B is zero.
(b) This case is similar to case (a). Hence, the answer is the same as(a).
(c) Torque `tau = IA xx B`
From the given figure, it can be observed that A is normal to the x-z plane and B is directed along the z-axis.
`:. tau = - 12 xx (50 xx 10^(-4))hatj xx 0.3 hatk`
`=- 1.8 xx 10^(-2)hatj Nm`
The torque is `1.8 xx 10^(-2)Nm` along the negative x direction and the force is zero.
(d) Magnitude of torque is given as:
`|tau| = IAB`
`= 12 xx 50 xx 10^(-4) xx 0.3`
`= 1.8 xx 10^(-2)Nm` ltbr. Torque is `1.8 xx 10^(-2)Nm` at an angle of `240^(@)` with positive x direction. The force is zero.
(e) Torque `tau = l vec(A) xx vec(B)`
`= (50 xx 10^(-4) xx 12) hatk xx 0.3 hatk = 0`
Hence, the torque is zero. The force is also zero.
(f) torque `tau = l vec(a) xx vec(B)`
`=(50 xx 10^(-4) xx 12) hatk xx 0.3 hatk =0`
Hence, the torque is zero. The force is also zero. In case (e), The direction of `I vec(A)` and `vec(B)` is the same and the angle between them is zero. If displaced, they come back to an equilibrium. hence, its equilibrium is stable.
Whereas, in case (f), the direction of `I vec(A)` and `vec(B)` is opposite. The angle between them is `180^(@)`. If disturbed, it does not come back to its original position. Hence, its equilibrium is unstable.