Fields of wires and coils in code

You will write three functions that give the magnetic field of a wire and of a coil. They use the Biot-Savart law.

The idea. A long wire has $B = \dfrac{\mu_0 I}{2\pi d}$. A wire of any length has $B = \dfrac{\mu_0 I}{4\pi d}(\sin\varphi_1 + \sin\varphi_2)$. A coil of $N$ turns has $B = \dfrac{\mu_0 N I R^2}{2(R^2 + x^2)^{3/2}}$ on its axis. Use SI units: metres, amperes and tesla.

Your tasks.

1. In field_long_wire, return the field of a very long wire at a distance d from it.
2. In field_short_wire, return the field of a straight wire at a point d away on the line that cuts it at right angles in the middle. The wire has a half length half. Use the two end angles, which are equal.
3. In field_on_axis, return the field on the axis of a coil of N turns, at a distance x from its centre. With x = 0 this is the field at the centre.