Take a steel core (K_m = 2500) electromagnet, bend it into a loop with a small air gap, and determine the B field in the gap. The cross-sectional area of the toroid is 4cm^2, and the air gap is 2.5mm. The current through the coil's 120 turns is 15 amps. The radius of the toroid is 7cm. Determine the B field in the air gap.
A full description of the problem, with a diagram, is here (MIT). It's problem 7.5.
Inside a complete toroid, B=mu_0*current*number of turns/2*pi*radius of the toroid.
The Attempt at a Solution
I would think the B field in a gap which is much smaller than the cross-sectional area of the toroid would be nearly the same as inside the toroid itself, which I calculated at 12.86 T.
But the given answer is far lower: 0.85 T. The answer can be seen here (MIT).
I don't understand the explanation. To me, it seems internally contradictory. It says that "the magnetic field strength in the gap will be approximately equal to the magnetic field strength in the steel," and then calculates the integral of B dot dl for the toroid and the gap in a way I haven't seen before. It seems to assume that you can put an Amperean surface in the air gap itself - which is not penetrated by any current - and get an answer to integral B dot dl that is non-zero.
It also says that B in the gap is nearly the same as B in the steel core, which by my calculations can't be true. B in the steel must be the 12.86 T that I calculated, right?
Can anyone help clear this up for me?