32: Interaction of a Current Loop with a Magnetic Field

INTRO: 
The effects due to the interaction of a current-carrying loop with a magnetic field have many applications, some as common as the electric motor. This problem illustrates the basic principles of this interaction.

Figure 1

Figure 2

Figure 3

Figure 4

Consider a current I that flows in a plane rectangular current loop with height a = 4.00 cm and horizontal sides b = 2.00 cm . (Figure 1) The loop is placed into a uniform magnetic field B⃗  in such a way that the sides of length a are perpendicular to B⃗ , and there is an angle θ between the sides of length b and B⃗ , as shown in the figures. (Figure 2)

PART A:

Will the interaction of the current through the loop with the magnetic field cause the loop to rotate?

  A) Yes, the net torque acting on the loop is negative and tends to rotate the loop in the direction of decreasing angle θ (clockwise).

  B) Yes, the net torque acting on the loop is positive and tends to rotate the loop in the direction of increasing angle θ (counterclockwise). 

  C) No, the net torque acting on the loop is zero and the loop is in equilibrium.

  D) No, the net force acting on the loop is zero and the loop is in equilibrium.

SOLUTION:
By using the right hand rule where the pointer finger indicates the magnetic field and the thumb represents the moving particle (aka direction of current), if you refer to figure 2, you can see that the force on both sides of the ring pull the ring out, increasing the angle. This observation is consistent with option B

PART B:

For parts B and C, the loop is initially positioned at θ=30°.

Assume that the current flowing into the loop is 0.500A . If the magnitude of the magnetic field is 0.300T , what is τ, the net torque about the vertical axis of the current loop due to the interaction of the current with the magnetic field?

Express your answer in newton-meters.

SOLUTION:

First off we have to determine the magnitude of the force on each vertical bar, which they will be inversely equivalent.
So, using the formula F = ILB, 
F = (0.5 A)(0.04 m)(0.3 T) = 6×10^-3 N
NOW, since the ring is skewed, we have to disregard the component of the force that is along the 'lever'. 
Using our handy right triangle, we can determine that the perpendicular component of the force will be
F₁ = F cos(θ) = 5.2×10^-3 N
Now, recalling from the beginning of physics, torque = F×l 
τ₁ = (5.2×10^-3 N) ⋅ (0.01 m) [we get this l from half the length of the top bar]
τ₁ = 5.2×10^-5 N⋅m
but this isn't our answer quite yet, remember they asked for the net torque. What we just found was the torque for the 'upper' half of the ring. To get our net torque, we must add the torque from the 'top' to the torque from the 'bottom'
luckily, the two torques are equal so we just multiply our previous value by 2 and determine that the net torque is 
1.04×10^-4 N⋅m

PART C:

What happens to the loop when it reaches the position for which θ=90∘, that is, when its horizontal sides of length b are perpendicular to B⃗  (see the figure)? (Figure 3)

A) The direction of rotation changes because the net torque acting on the loop causes the loop to rotate in a clockwise direction.

 B) The net torque acting on the loop is zero, but the loop continues to rotate in a counterclockwise direction.

 C) The net torque acting on the loop is zero; therefore it stops rotating.

 D) The net force acting on the loop is zero, so the loop must be in equilibrium.

SOLUTION:

so once the ring rotates to the 90° point, there will be no more force on the edges of it, however, there won't be any forces on the other side either, resulting in it to continue rotating until something stops it, which is satisfied by B).

PART D:

Now suppose that you change the initial angular position of the loop relative to B⃗ , and assume that the loop is placed in such a way that initially the angle between the sides of length b and B⃗  is θ=120, as shown in the figure. (Figure 4) Will the interaction of the current through the loop with the magnetic field cause the loop to rotate?

A) Yes, the net torque acting on the loop is negative and tends to rotate the loop in the direction of decreasing angle θ (clockwise).

B) Yes, the net torque acting on the loop is positive and tends to rotate the loop in the direction of increasing angle θ (counterclockwise).

C) No, the net torque acting on the loop is zero and the loop is in equilibrium.

D) No, the net force acting on the loop is zero and the loop is in equilibrium.

SOLUTION:

Same idea as in the first problem, except in the opposite direction. This means that there will be torque on the ring just in the other way now. So, A).