What Is the Equilibrium Angle of a Current Loop in a Magnetic Field?

[futurepocket]

Homework Statement

A square loop of wire with side 0.60 m carries a current of 9.0 A as shown in the figure (http://img847.imageshack.us/i/springquestion.jpg/). When there is no applied magnetic field, the plane of the loop is horizontal and the nonconducting, nonmagnetic spring (k = 550 N/m) is unstretched. A horizontal magnetic field of magnitude 1.3 T is now applied. At what angle theta is the wire loop's new equilibrium position? Assume the spring remains vertical because theta is small. (HINT: Set the sum of the torques from the spring and the magnetic field equal to 0)

Homework Equations

tau = NIABsin(theta)
tau = rFsin(theta)

The Attempt at a Solution

Well, the hint says it all, except I can't seem to figure out the torque from the spring. I assume that since F = -kx, we put that in for the force for the torque, the 'r' is the distance from the axis of rotation to the spring (which I am assuming is 0.6 / 2 since it says its a square but the image doesn't look like a square). Unfortunately, I don't get the right answer when I do that because the sin(theta) can just cancel out on both sides.

Therefore, I considered that we assume the spring stays vertical, so sin(theta) for the torque from the spring doesn't exist.

So we get:

-rkx + NIABsin(theta) = 0

but we don't know 'x', the stretch of the spring. So I used Newton's laws, and summed up the forces and figured out that the force due to spring must be equal to the magnetic force on the spring... so I did that, and still didn't get the right answer. I am totally lost, any help?

 

[anathema]

Thank you for your post. It seems like you are on the right track with your attempt at a solution. Here are a few suggestions that may help you to arrive at the correct answer:

1. Remember that torque is a vector quantity and it has both magnitude and direction. When you write the equation for torque, make sure you include the direction of the torque. In this case, the torque from the spring will have a negative sign since it is acting in the opposite direction of the magnetic torque.

2. The distance 'r' in the torque equation is the distance from the axis of rotation to the point where the force is applied. In this case, it is not the distance from the axis to the spring, but rather the distance from the axis to the point where the spring is attached to the loop. You can use the Pythagorean theorem to calculate this distance.

3. To find the stretch of the spring, you can use Hooke's law, F = -kx, where 'F' is the force from the spring, 'k' is the spring constant, and 'x' is the stretch of the spring. You already know the value of 'k' and you can calculate the force from the spring using the equation you mentioned in your post (since you know the value of the magnetic force). This will give you the value of 'x', which you can then use to calculate the angle theta.

I hope this helps. Good luck with your problem!