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Problem number five.
28-57.
The torque on a current loop.
Let's set up again a
right-handed coordinate system.
Always right-handed.
'x y z'.
And we have a loop, rectangular loop, which comes
off at an angle and the angle is theta.
It's hinging here, frictionless about the z-axis.
This length is 'l1', and this length is 'l2', this is also
'l2' and this is also 'l1'.
And somehow, for reasons that I have not explained, there's
a current flowing in this directions, the same in this
direction, in this direction, and in this direction.
This current is 'i'.
We know the total my mass of this whole thing because you
are being told that the mass for unit length is 0.19 grams
per centimeter.
You know 'l1' and 'l2', so you can calculate the mass or this
whole thing.
The force on each one of these individual wires in the
presence of a magnetic field would be 'i'
times 'l cross B'.
And 'B' is given is in the '+y' direction.
Uniformly throughout.
If you analyze the forces on each one of these four
elements, and I would like you to do that on your own, by
carefully evaluating the direction of 'l cross B'.
'I' is always a little element in the
direction of the current.
So 'l dl' would be here.
And the 'dl' would be there.
Then you should be able to convince yourself that the
force at the bottom here, this part here,
is in the 'x' direction.
I'll call it 'F of x'.
And therefore right here, it's the same in magnet I call it
'- x', just to tell you in which direction it is.
This wire here experiences a force in the '+ z' direction,
and this one experiences a force in the '- z' direction,
and the two exactly cancel each other out.
So I don't have to worry about these two.
But these two do not cancel each other out.
They do cancel each other out in terms of the sum of all
forces being 0, but not the sum of all torques being 0.
So there's a torque on this system.
But yet we know that it is an equilibrium.
So the only way that it can be an equilibrium that there is
not a torque, this is a counter- clockwise torque, it
has to be a clockwise torque to hold it in equilibrium.
I'm going to now make a drawing from the side.
I call this hinge points through which is z-axis goes
'P', this one is 'l1', this angle is theta and I'm going
to draw all the forces in there that I can think of.
This is the total mass off the loop 'mg', here is that force
'Fx' due to the magnetic fields and the current.
Well there must be here this force '-x', right?
I call it 'F minus x', it's the same in magnitude as this
one, they cancel each other out.
But since the sum of all forces must be 0, according to
Newton, and this hinge point, there must also be
a force 'mg' up.
So you'll see now that the sum of all forces is now 0.
But know the sum of all torques is also 0.
It doesn't matter which point you take, whether you take
this point, this point, this point, this point, or this
point, or this point, or this point, the sum of all
torques will be 0.
I might as well choose this points 'P'.
So I will calculate now for you the torque relative to
point 'P' which must be 0.
When I am here, there is no contribution to this force and
this force.
I have a clockwise contribution due to this one.
So there is this arm times this force.
Now the arm has a length 0.5 'l1' times the sine of theta
times 'mg'.
But that is the clockwise one.
I called this clockwise, must be equal to the
counter-clockwise, I give them both a positive sign.
And the counter-clockwise is relative to this point 'P',
this distance times this force.
And this distance is ''l1' cosine theta' times 'Fx'.
Notice I have one equation with one
unknown and that is 'Fx'.
Because I know all the other things. 'l1'
cancels, by the way.
I know say theta.
But I also know that 'Fx' equals 'i' times 'l2', mind
you it is the length of this wire here perpendicular to
'l1', times that magnetic field 'B', which is only in
the y-direction.
I know the magnetic fields, I know 'l2', I know--
I know 'Fx', I know 'i' and I know 'l2', so out pops the
magnetic field.
So I can calculate the magnetic field.
Interesting problem.
So due to this magnetic field, this loop will come out, it
will experience a torque, but the gravitational torque
exactly balances it, and so it will be standing at
equilibrium at this angle.
You can argue that if you increase the current the angle
will probably become larger, and if you increase the
magnetic fields the angle will probably also become larger.