Engineering Electromechanical energy conversion problem

[cnh1995]

Homework Statement

The problem statement is as follows:

Relevant Equations

Energy stored in a magnetic circuit=
E=(1/2)*Flux linkage*Current

Force on a movable part in the magnetic circuit=
F= -dE/dx.

Here, I have assumed very quick movement of the moving part. So, as per the mmf-flux curve, flux linking with the system will be unchanged during the movement. To support this assumption, I have made use of the following analysis.

In this graph, the movement of the part happens quickly (small air gap), hence, the the flux linkage will remain unchanged (line PQ) throughout this small interval.

I hope I have sufficiently presented my attempt at a solution. Sorry for the messy white-board though.

The answer I got is F=282.4N but the answer according to the official answer key is 188N.

Where did I go wrong here? Any help is appreciated.

 

[berkeman]

It looks like some information is missing in this problem, IMO. A 10cm air gap is giant, unless the cross-sectional area of the magnetic core is also huge. What are the rest of the dimensions for this system? What are the dimensions of the metal piece in the air gap? Is it magnetized?

If it's not magnetized, then it will be the non-uniformity of the flux in the air gap that will attract the moving metal piece to one of the core surfaces at the air gap (whichever is closer when the metal piece is released), no?

Also, please use LaTeX instead of the pics of the whiteboard

https://www.etcourse.com/news-blog/air-gap-in-magnetic-circuits

 

[cnh1995]

I found this article https://www.google.com/url?sa=t&sou...sQFnoECBYQAQ&usg=AOvVaw2PFRA8e32CweCuuMzO6JaC.

The force applied on the moving part at t=0+ here will be simply the derivative of field energy w.r.t displacement of the moving part, with constant current.
Using this insight, I got F to be 141.4 N.
I will post my work in some time.

Now I doubt if the official answer key is correct.