(a)A long metal cylinder of radius a has the z-axis as its axis of symmetry.The cylinder carries a steady current of uniform current density J = Jzez. Derive an expression for the magnetic field at distance r from the axis,where r<a. By resolving the cylindrical unit vector eφ along the x-and y-axes, show that the magnetic field at any point P inside the cylinder is
B(x, y, z)= μ2 0 Jz (−yex + xey) ,
where P has Cartesian coordinates (x, y, z), and(x2 + y2) <a2.(15 marks)
eφ = -sin(theta) + cos (theta)
The Attempt at a Solution
I know that the above equation is relevant but I am not sure how that this resolves into
(−yex + xey).
I would have thought that the only way this works is if sin(90) = 1 and cos (0) = 1 then converting this back to cartesian components would give the required answer.