Force between two opposite faces of the core across an air gap: In a magnetic circuit involving air gap, magnetic force will exist between the parallel faces across the air gap of length x. The situation is shown for a magnetic circuit in figure A (i). Direction of the lines of forces will be in the clockwise direction and the left face will become a north pole and the right face will become a south pole. Naturally there will be force of attraction Fa between the faces. Except for the fact that this force will develop stress in the core, no physical movement is possible as the structure is rigid.

fig(A) Force between parallel faces

Easiest way to derive expression for F_a is to apply law of conservation of energy by using the concept of virtual work. To do this, let us imagine that right face belongs to a freely moving structure with initial gap x as in figure B(ii). At this gap x, we have find F_a. Obviously if we want to displace the moving structure by an elemental distance dx to the right, we have apply a force F_e toward right. As dx is very small tending to 0, we can assume B to remain unchanged. The magnitude of this external force F_e has to be same as the prevailing force of attraction Fa between the faces. Where does the energy expended by the external agency go? It will go to increase the energy stored in the gap as its volume increase by A dx. Figure A (iii) shows an expanded view of the gap portion for clarity. Let us put it in mathematical steps as follows: