I Does a Railgun's Current Violate Conservation of Momentum?

[renormalize]

Thank you for introducing this useful and informative calculation process.

However, I noticed that doing this experiment can be very difficult and expensive.

Agreed, my idealized device is hardly practical. So let's make a more realistic example by increasing the total mass and storing energy in a commercial capacitor. For example, on Amazon you can find this $1\text{ gm}$ Torong HV ceramic chip capacitor:

characterized by $C=10\text{ nF},V=30\text{ kV}$. Coupling this to a lightweight antenna structure, we could perhaps put together a device of, say, $M=50\text{ gm}$ total. Energy-momentum conservation then yields:$$E_{\text{rad}}\approx E_{\text{cap}}=4.5\text{ J},\:Mc^{2}=4.5\times10^{15}\text{ J},\:v=300\text{ nm/s},\:\frac{1}{2}Mv^{2}=2.25\times10^{-15}\text{ J}$$So the recoil speed $v$ is only on the order of hundreds of nanometers per second, and this is an upper limit due to resistive losses and the reduced directivity of a real antenna. Nevertheless, I can imagine that a sensitive lab experiment in vacuum might be able to detect the motion.

 

[alan123hk]

So the speed v is only on the order of hundreds of nanometers per second, and this is an upper limit due to resistive losses and the reduced directivity of a real antenna. Nevertheless, I can imagine that a sensitive lab experiment in vacuum might be able to detect the motion.

Agree that this movement should be detectable with advanced scientific instruments.