The magnetic field outside a long straight wire carrying a steady current I is...

The magnetic field outside a long straight wire carrying a steady current I is

The electric field inside the wire is uniform:

where ρ is the resistivity and a is the radius (see Exs. 7.1 and 7.3). Question: What is the electric field outside the wire?29 The answer depends on how you complete the circuit. Suppose the current returns along a perfectly conducting grounded coaxial cylinder of radius b (Fig. 7.52). In the region a < s < b, the potential V (s, z) satisfies Laplace’s equation, with the boundary conditions

This does not suffice to determine the answer—we still need to specify boundary conditions at the two ends (though for a long wire it shouldn’t matter much). In the literature, it is customary to sweep this ambiguity under the rug by simply stipulating that V(s, z) is proportional to z: V(s, z) = z f (s). On this assumption:

(a) Determine f (s).

(b) Find E(s, z).

(c) Calculate the surface charge density σ(z) on the wire

Reference example 7.1

A cylindrical resistor of cross-sectional area A and length L is made from material with conductivity σ. (See Fig. 7.1; as indicated, the cross section need not be circular, but I do assume it is the same all the way down.) If we stipulate that the potential is constant over each end, and the potential difference between the ends is V, what current flows?