How Large Is the Force on a Rod Partially Inserted in a Solenoid?

[zzachattack2]

Homework Statement

A long cylindrical rod (permeability u , length L0, radius R) is partially inserted into a long
solenoid (length L1, radius R, turn density n, and fixed current I)

How large is the force on the rod? Our hint is to first calculate the total energy in the field.

Homework Equations

W= 1/2*L*I^2 for calculating energy and finding force from energy, F = dW/dL

The Attempt at a Solution

So my attempt to solve this was of course first find the energy stored in the field. Of course the Inductance of the solenoid is just L=u0*n^2*pi*R^2*L1 and this can just be plugged into the formula above for finding total energy in the field. My guess is that you should find how the total magnetic field changes by inserting a permeable core, and thus how the energy in the field changes. This difference can be used to calculate the force on the rod. However I'm having a lot of trouble finding the difference in energy when the rod is "partially" inserted. Any ideas? Thanks.

 

[Jhero]

Dear fellow scientist,

Thank you for your forum post. Your approach to calculate the total energy in the field is a good starting point. However, in order to find the force on the rod, we need to consider the change in energy as the rod is inserted into the solenoid.

To do this, we can use the formula for the inductance of a solenoid with a permeable core, which is L = u*n^2*pi*R^2*L1 + u0*(L0-L1). Here, u is the permeability of the core material and u0 is the permeability of free space. The first term represents the inductance of the solenoid with the core fully inserted, while the second term represents the inductance of the solenoid with no core inserted.

Next, we can use the formula for the total energy in the field, W = 1/2*L*I^2, to find the energy with the core fully inserted and with no core inserted. The difference between these two energies will give us the change in energy as the rod is partially inserted.

Finally, we can use the formula F = dW/dL to find the force on the rod. This will give us the force on the rod as it is inserted into the solenoid.

I hope this helps in solving the problem. If you have any further questions, please let me know. Good luck with your calculations!