# What is the current in a short linear antenna in an alternating $E$-field?

Consider a short linear wire with some low but non-zero resistance, such as a copper wire. If this wire is then placed in an alternating electric field that is oriented parallel to the wire, the field will push the free charges in the wire back and forth along the wire, creating an oscillating current in the wire. I'd like to know what this current is. Essentially, this problem considers transfer of energy from electromagnetic radiation to an antenna.

I can start on the problem some. The external $$E$$-field is $$E(t)=E_0 \sin(\omega t)$$ I think the current must be $$I(x,t) = E_0 \sin(\omega t+\delta)\sum_{n=1}^\infty A_n \sin\left(\frac{n \pi x}{L}\right)$$ where the first portion shows that it has the same time dependence as the electric field, but with some phase lag and the second portion shows that it can be decomposed into a Fourier sine series. The terms in this series obey the boundary conditions that the current equals zero at the ends of the wire. The $$A_n$$ values are unknown (to me) and need to be found.

It seems to me that this should be a standard equation in the theory of antennas but I can't seem to find it. All I can find is the radiation field produced by a center-driven linear antenna (e.g. https://www.cv.nrao.edu/course/astr534/AntennaTheory.html), and I'm told that the reciprocity theorem allows me to reverse the problem to get the current from the field, but I'm not sure how to do this.

As an aside, there are going to be three regimes to this problem, for the driving frequency much slower than resonance, much faster than resonance, and in the vicinity of resonance. Ideally, I'd like all three answers. Any suggestions will be much appreciated!

• This is really an EEng question. Have you checked there? – ZeroTheHero May 13 at 19:26
• Why do you think the current at the boundaries should be zero? Shouldn't the spatially uniform external field excite a spatially uniform current in the rod? – wcc May 13 at 20:27
• @ZeroTheHero, I'm not sure this is a strictly engineering problem. At least you learn about dipole antennae in standard E&M physics curriculum, and some of the questions are geared toward broad physics concepts (resonance etc.). – wcc May 13 at 20:34
• @AmIAStudent The current at the ends of the wire is zero because electrons can't flow beyond those points. At high driving frequency, I suspect that you're mostly right that the external field would drive a uniform current in the rod, although it still has to decrease to zero at the ends. At low frequencies, the current creates a charge distribution in the rod that then acts against the external electric field, which reduces the current. This happens more near the ends than the middle, creating a non-uniform current in the rod. – user2419194 May 14 at 21:05
• "The current at the ends of the wire is zero because electrons can't flow beyond those points." Yes, electrons cannot flow beyond and they accumulate. In the antenna, we are exciting a dipole moment so there will be time-dependent charge accumulation at the ends of the rod (e.g. + on one end and - on the other). So the current doesn't have to be zero. – wcc May 14 at 21:30