alanf said:
I am still very puzzled by one thing. If the solenoid was simply straight, and not curved into a toroid, B at the center would be 12.86 T, right? If so, I'm really surprised that curving it somehow reduces B in the solenoid itself, and then gradually brings B back up to 12.86 T as the gap is closed. I feel like I'm fundamentally misunderstanding the physics here. I keep looking for some way to put a current or a changing E field through the gap, but of course there is none.
I believe it is true that bending a solenoid with a steel core to form a toroid with a gap will cause B inside the steel to first decrease and then to increase as the gap is closed. I think you can make this plausible in the following way.
Apply Ampere’s law in the form ##\oint \frac{\vec{B}}{\mu} \cdot d\vec{l} = I_{enc}## to a long straight solenoid with a steel core where the path of integration passes through the entire length of the solenoid and then returns outside the solenoid to form a closed path. Then we get $$\frac{B_s L_s}{\mu} + \frac{B_a L_a}{\mu_0} = NI$$ where N is the number of turns of winding of the solenoid, ##I## is the current in winding, ##L_s## is the length of the steel core, ##B_s## is the average field along the part of the path inside the steel, ##\mu## is the permeability of the steel, ##L_a## is the length of the path in the air, and ##B_a## is the average field along the path in the air.
The first term on the left tends to be small due to the large value of ##\mu##. The second term tends to be small due to the fact that in the air the B field spreads out and becomes weak. For a long, thin solenoid the average field, ##B_a##, along the part of the path in the air becomes so small that the first term dominates. Solving for ##B_s## gives ##B_s \approx \frac{\mu NI}{L_s}##.
Now suppose the ends of the solenoid are bent around to form part of a toroid with a large air gap. ##L_a## decreases while ##B_a## increases in the gap due to less spreading (fringing) of the field in the gap. As the gap is made smaller, ##B_a## actually increases faster than ##L_a## decreases and the second term on the left of the equation starts to become significant. Since the right hand side remains constant, you can see that ##B_s## must decrease as the gap is made smaller.
When the gap gets small enough so that fringing of B is negligible in the gap, you can say ##B_s \approx B_a##. Then the equation leads to ##B_s \approx \frac{\mu N I}{L_s+ \frac{\mu}{\mu_0}L_a}##. This is essentially the equation used in the problem (note ##\mu = \mu_0 \kappa_m##).
If the gap ##L_a## is made extremely small, then ## \frac{\mu}{\mu_0}L_a## begins to become negligible compared to ##L_s##. Then you see that ##B_s \approx \frac{\mu NI}{L_s}## which is the same expression we had for the long, straight solenoid. So by the time the gap is completely closed, ##B_s## has increased back to its original value in the straight solenoid.
It is important to keep in mind that ##\mu## is not a constant for ferromagnetic materials. It depends on the value of B and even the history of the variation of B in the steel. So, while the gap is changing, ##\mu## will also be changing. The value of ##\mu## when there is a 2.5 mm gap might not be the same as the value of ##\mu## in the long straight solenoid or in the toroid without a gap.
