Question

Consider a cylindrical wire of radius R (indefinitely long) that carries a total steady current I such that there is a constant current density j across the profile of the wire (for the first part of this task, consider just a current density in vacuum)

a) in order to calculate the magnetic induction it is suitable to work in cylindrical coordinates. Considering Boundary conditions at ρ→∞, the magnetic induction ca be written as B=B_ρ (ρ,φ,z) e_ ρ + B_ φ(ρ,φ,z)e_ φ Use the symmetry of the infinitely long cylindrical wire(and the corresponding current density) to simplify this ansatz for the magnetic induction.

b) use maxwell´s equation (no magnetic monopoles) to show that there is no component of the magnetic induction in the radial direction. Hint: use Gauss theorem together with a cylindrical volume.

c) How does the result of your calculation change if only the conductor of the wire has a magnetic susceptiblity χ>0? Hint: use the material equations and the equation for linear media.

Answers to the same problem have been posted but please DO NOT copy them. Thank you very much in advance.

Answer 1

For a magnetic field to be produced, there should either be a current flowing or there should be a changing electric field.

In this problem, there is only a current flowing in a certain direction that is constant in magnitude as well.

Assume the current is flowing in the z direction. Use cylindrical coordinates.(r,theta,z)

Since there is only a current flowing in the z direction, using the right hand grip rule, there will be a magnetic field that will circle counter clockwise around the current.

There is no doubt about the presence of any other magnetic field components. Magnetic fields are produced by some cause and this cause is the electric cirrent flowing in the z direction. There cannot be an arbitrary radial component of the magnetic field just appearing by magic.

For the magnetic induction , consider a circlular Amperian loop whose axis coincides with that of the wire.

If the radius of the circle is larger than the wire,

and the magnetic field is along the tangential direction.

If the radius of the circle is less than the wire,