Understanding Torque on a Solenoid: An Explanation and Analysis

[schaefera]

I'm looking at this picture from part of an explanation of the torque on a coil of current arranged as a solenoid: http://www.physics.sjsu.edu/becker/physics51/images/28_30_Torgue_on_solenoid.jpg

Now, they claim that the torque is directed into the page, but I can't understand why. I agree that the torque of an individual current loop about its center would be into the page, but if we're trying to find the torque about the center of the entire solenoid, I seem to think that the total torques should cancel out.

Here's my reasoning. The force on the left side of the wires will point up, and the force on the right side points down, in accordance with the right hand rule. Now, getting the torque on, say, the top left corner, we find that it points out of the page due to τ= r x F and by the same equation, the torque due to the top right corner of the solenoid points into the page. This would happen for each loop in the solenoid, so shouldn't the torques cancel? Why can we treat the entire solenoid as a single current loop, and multiply by the of turns in it?

 

[Simon Bridge]

I seem to think that the total torques should cancel out.

You can tell the torques should not cancel out because the solenoid acts as a magnet - and you know that a magnet will rotate in an external magnetic field.
On top of this, we can actually put an actual solenoid into a magnetic field and measure the net torque ... so your argument is saying that your current understanding of the theory predicts something that is contradicted by experiment.

Here's my reasoning. The force on the left side of the wires will point up, and the force on the right side points down, in accordance with the right hand rule. Now, getting the torque on, say, the top left corner, we find that it points out of the page due to τ= r x F and by the same equation, the torque due to the top right corner of the solenoid points into the page. This would happen for each loop in the solenoid, so shouldn't the torques cancel?

If you draw the vectors, you'll find the "top left" torque is a different magnitude to the "top right" torque. They don't cancel. The right-hand one is bigger than the left-hand one.
(Even though the forces are the same, they are at different angles to the moment arm.)

Take another look at the torque from the element of loop top-left ... the force points up, the angle means the torque is slightly anti-clock.

Now look at the bottom right element ... the force is pointing down, the angles are the same but complimentary, so it produces the same magnitude torque, and in the same direction ... these two elements create a couple. So they add together.

Now sum all the couples up and down the solenoid.

 

[Philip Wood]

The torque on each turn is clockwise as we look at the diagram, that is the torque vector, G, for each turn is into the page. I think we agree over this.

But the particular torque experienced by each turn is due to a couple, equivalent to a pair of equal and opposite forces not in the same straight line. It's easy to show that the torque due to a couple is the same about any point. So torques due to individual turns can simply be added, without worrying about what point the torques are taken about. The resultant torque vector for an N turn coil is therefore simply NG, even if the turns are 'staggered' over a distance.