Point of action of magnetic force on a current loop

[sergioro]

Hello everyone,

Translation of extended objects is described taking the
net force acting on the center of mass of the extended object.
But to compute rotational motion, one needs to considers
each force on their point of action.

For example, let's consider a current I flowing in a loop consisting of
a conductor forming a semicircle and another as a straight segment
trough the diameter of the semicircle. Assume the current flows
counterclockwise in the loop which lies on the XY plane (being the semicircle part on the
+Y-axis) and that a constant magnetic field in the +X direction is acting on the loop.

In this situation a torque will make the loop to rotate around the
Y axis.

How can one prove that the forces responsible of the torque, one acts
at the mid point of the piece of the curved loop in the first
quadrant and the other at the mid point of the piece of the curved loop
in the second quadrant?

Thanks in advance,

Sergio

 

[Simon Bridge]

The fact there is a measurable torque is the proof of the force distribution you talk about ... but you should think of the force as distributed around the entire loop - not acting at a single point.

Mathematically you can show this is consistent with current models by dividing the loop into very short segments which can be treated as if they are straight, much like we often treat the ground as flat, and then applying the rule for a current in a straight wire for each segment... find the relationship between the position of the segment in the loop with the force on the section.

We can replace the distributed force by an equivalent couple that works like you describe.

 

[sergioro]

Thanks, Simon. The first part of your comment triggers the intuition I was missing.

 

[Simon Bridge]

NO worries - that's what I'm here for.