Question

If a solid conducting cylinder was quickly inserted in a hole of the current-carrying pipe, it will not produce eddy currents because the field inside is zero. However, there is a magnetic in the outside of the current-carrying pipe, therefore, at entry/exist there is a magnetic field and the solid cylinder will experience Eddy currents on the surface.

Draw the following:

- The system

- The Eddy currents

- The direction of velocity of the solid cylinder(at entry or exist)

- The direction of opposing force, would it effect the entry/exist velocity?

Please clarify with your answer.

Answer 1

This a a confusing question. So, I am making some assumptions.

Firstly, the hollow pipe. I am assuming it is carrying current in the axial direction (which sounds a bit practical)

(current direction: axial and directed away from the small solid)

If this is the case, the magnetic field outside the pipe will be a tangential magnetic field that varies inversely with distance from the axis of the pipe.

The magnetic field inside the pipe is zero because of no current being "inside" the pipe all current is on the surface of the pipe.

Now, the next assumption: For us to study the magnetic field just by intuition (as per this problem), we need a simple tangential magnetic field (as opposed to weird looking magnetic fields that looks like a large pile of crumpled rubber bands or multiple worms that somehow form closed loops etc.) . This can be generated only under the ideal circumstance that the pipe has infinite length.

In this case, the pipe apparently does not have infinite length because whe are able to find one end and are trying to put a cylinder inside ( yes. it should be symmetrically infinite (i.e. in both directions to study a nice uniform field)

That being said, I will now change the problem a little bit and answer the question of:

" Draw the direction of the eddy currrents and state if opposing force affects the velocity of a solid sylinder that is exiting a region of uniform tangential magnetic field (that decays inversly with distance) and going to a region of zero magnetic field"

Solution:

Initially, the solid is exposed to this uniform tangential magnetic field. When it is completly inside, it is exposed to zero magnetic field. But, when it is partly in, that is interesting.

The solid cylinder:

   The velocity direction is axial.

   The magnetic field is tangential.

Consider this picture, where I draw only the solid cylinder, sliced along its length.

I have drawn the magnetic field that is linked with a surface element on the lengthwise cross section.

Now, if this magnetic field is made to dissapear slowly, then, by lenz's law, currents will be set up so as to increase currents in that derection.

They will look like this:

These currents are planar, with the plane parallel to the axis of the solid cylinder.

But I have considered only one plane! there are infinite such planes. Eg:

Now, the entry velocity is opposed by the created forces. This is simply because the velocity of the solid towards the zero magnetic region is the cause that is changing the magnetic flux. By lenz's law , forces are set up precisely to oppose this change! So, without considering any gemetry or vectors or things like that, we simply say "yes. the opposing forces will try to stop the moving solid cylinder"