Fields From Charge and Current Distributions

Good day I wanted to ask about a concept of Ampere's and Gauss's law. This professor said that suppose there is a thick cable of radius R carrying a current I. We want to find the magnetic field B at a distance r. To do this we will take a circular Amperian surface and apply Ampere's Law int(B.dl) = uI B(2pir) = uI B = uI/2pir This result is the same as that for a thin wire at the centre of the Amperian surface. My question is: Can we apply Ampere's Law if the cable has a current that is different in different parts of the wire? For example, what is there is a continuously increasing current from the left to the right of the cable(current on the left is I and on the right is 2I). How will we calculate the B then? Will we integrate to find the total I, and THEN put it in the Ampere's Law? Can this cable (with varying I) still be assumed as a thin wire at the centre of the surface, even though now the current is more towards the right of the Amperian surface than its left? Secondly, does this principle apply to Gauss's Law as well? What if there is a non-conducting sphere, radius R, enclosed by a Gaussian surface, radius r, and the sphere has a continuously increasing charge density from left to right. Can we still apply Gauss's Law, and can we still assume it as a point charge in the centre of the Gaussian surface? I apologise for the long question.
- Daniyal Ahmed (age 19)
Karachi, Pakistan

That's a very thoughtful question. Far away from the wire, the field can be calculated treating the current as flowing along a line in the middle of the wire. Close to the wire, the distribution matters. In the middle of the wire, for example, there's zero field if the current is symmetrical around that point but there is a field if the currents on each side are different.

The same principle applies to electric fields coming from charge distributions. 

As you guessed, the fields can be calculated by doing integrals over all source regions of the current or charge densities multipled by the appropriate geometrical factors for how far away that is in what direction.

The fields well outside each source region can be also expressed as the sum of monople, dipole, quadrupole, etc. components. The monopole piece is zero for magnetic fields, at least for all known sources. The other pieces depend on how the charges or currents are arranged.

Mike W.

(published on 11/04/2014)