# Field created by a wire, known voltage

I am having some mental troubles with this problem I came across reading about transmission lines (I mean, it is not a problem from a book, I just thought about it).

Say I have a couple of wires (parallel infinite straight lines) with known electric potential (in free space). They model a transmission line and they carry some AC current (the relation between voltage and current is given by some constant, the characteristic impedance).

I want to know the Electric field (the magnetic part doesn't bother me) created by it outside the wire. Is the problem well posed with just this information? If no, exactly why? If yes, how can I find the solution? In the latter case I don't expect a closed form solution, but the procedure to find the equations, or the equations themselves. I think it would require numerical computation to solve them.

Edit: how about the case with constant voltage? Would it be just zero?

• There is a closed form for this, for both AC and DC cases. You generally will need to care about the magnetic part too, for a time varying problem. Commented Jan 8, 2014 at 19:46
• Are you familiar with the integral forms of Maxwell's equations? Commented Jan 8, 2014 at 19:51
• Yes. In this case, though, as transmission power lines oscillate at 50-60Hz, there's little electromagnetic interaction. Do you have a link or reference for that closed form? Commented Jan 8, 2014 at 21:44
• For quasistatic approximations, you can use the integral forms. An alternative is to treat this as full-wave EM problem in 2D (I'll take some time and write a detailed answer later). Outline: Each wire has a current density, with an associated magnetic vector potential that will take the form of a Hankel function as a distance from that wire. The total magnetic vector potential will be the superposition of the two potentials. The magnetic and electric fields derive from the potential by taking some vector derivatives. Then you can see what terms are small because of the low frequency. Commented Jan 9, 2014 at 17:14
• After doing that and using the approximations for small frequency, I would basically expect the magnetic field to look like the DC magnetic field of two currents, modulated by an AC wiggle, and the electric field to look like that of two infinite line charges, modulated by an AC wiggle. But I'll verify and post later. Commented Jan 9, 2014 at 17:16

You can measure the magnetic field due to a current carrying wire (as you say you are comfortable with it) so, in case of an alternating current the oscillations of current produce a time dependent magnetic field and such magnetic fields have a property of inducing electric fields. It can be mathematically calculated as $E = -\frac{d\Phi_B}{dt}$