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Assume an inifinitely long, straight thin electrical conductor with current I.
It follows from Ampere's law that the B-field intensity is proportional to 1/r, where r is the distance from the conductor This would make the B-field tend to infinity for small r.
Does this mean that to model the B-field around a real physical conductor, it is really necessary to factor in the thickness of the conductor? (I'm thinking that the problem only occurs for infinitely thin conductors.)
Indeed, an infinitely long and thin wire with a current $I$ has a magnetic field given by (in SI units):
$$B = \frac{\mu_0 I}{ 2\pi r}$$
Now suppose your wire has a finite radius. Ampère's law shows that as long as cylindrical symmetry is mantained, the field depends only on the current and not on the detailed properties such as the wire's radius or the current density as a function of radius. In other words, the above formula is still valid outside the conductor. Therefore, when considering a wire that has a finite width, if we only care about the field outside we don't need to care about how the wire looks. This is analogous to the theorem that says that the electric field outside a spherical charge distribution is given by $Q/4\pi\epsilon_0 r^2$, where $Q$ is the total charge.
