Right then. Getting back to intent of OP re coil gun. A 'serious' coil gun will actually be a rail-gun and won't be using magnetic media but typically capacitor discharge through a net single-turn circuit. Quite some years back I got talking to a lecturer at a local Uni who had been asked by a government funded rail-gun researcher whether to use the standard Biot-Savart expression, or a long discarded alternative owing to variously Weber and Ampere and Neumann being championed at the time by a Peter Graneau & co. I was surprised there was any real uncertainty. Pointed out that Weber/Ampere/Neumann formula that predicted longitudinal stresses in a current carrying wire was easily discounted just on a simple scaling argument. Not sure if that ever got back to the researcher, or whether that particular project ever went anywhere useful. Apparently courtesy of BAE the US navy now has a useful 'killer' version.