I was going to conduct this experiment myself, but I decided to ask it. Picture a balance scale. On each pan sits a spool of insulated wire. The spools are identical. One spool is connected to an alternating current source. Adjustments are made to insure that the scales are in balance. A switch is thrown that sends a current through the coil. What happens? If the pan with the current moves down could we then assign a weight to the field? Note: The scales and the table are plastic.
Answer
For simplicity, let me take two capacitors instead of coils of wires (this way, I don't have to deal with induction from the solenoid). Though I have changed the system slightly, the basic idea is the same: if I have two identical capacitors on either end of a plastic scale, and I suddenly charge one capacitor, will the scale register this change? If so, does this mean that the electric field has mass?
We know that the capacitor will gain some energy given by $E=\frac{1}{2}CV^2$. By Einstein's $E=mc^2$, the capacitor will also gain some mass. However, it will be such a small amount that you might not be able to measure it. Now, the question remains: does the field gain mass? The answer is nicely summarized on Quora, but with magnetic fields. Basically, even though the electromagnetic field harbors massless excitations, it can still gravitate. So, in a sense, you can say that the field has some mass, but I think it is safer to say that the energy imparted to the charged device gives said device some additional mass (from $E=mc^2$) rather than the field itself gaining mass.
You might also be interested in Bohr and Einstein's debate about the light box. Here, a photon escapes from a box and changes the box's weight as a result (sort of the opposite of what I described above, where we have a gain in weight). This thought experiment seemingly violates the energy-time uncertainty principle, though Bohr was able to counteract it by using an argument from general relativity.
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