How would I find the vector potential, A, of a uniform magnetic field?

Answer 1

The vector potential is defined only to within the gradient of a scalar. That is, you can add the gradient of any scalar field to the answer below and get an equally valid expression for the vector potential.

If B is the constant (vector) value of the magnetic field then the vector potential is:

A(r) = -r x (B/2)

where r is the the position vector and x means cross product. In Cartesian coordinates in which the magnetic field is pointing in the direction of the z axis, the three components of A are:

A_x = -(B/2)y
A_y = (B/2)x
A_z = 0

where B here is a constant number.