The time constant is 25ns, not 100ns, so the initial slope is 220V/25ns = 8.8Gv/s

There is about 100nC in the capacitor which has to drain on timescale of 25ns, ie about 4A,

and it has to rise to that sort of value in that timescale roughly, 4A / 25ns ~ several 100MA/s

I used the voltage as a function of time equation for a capacitor.

Tau will be equal to RC, where R is simply the mosfet Ron and the impedance of the transducer, 50ohm, in series.

I found a typical 250V mosfet that had an Ron of 0.32ohms, so I used R = 50.32ohm.

Vc(t) = Vo * e^(-t/tau)

Vc(25ns) = 76.44V

Vc(50ns) = 26.56V

Vc(75ns) = 9.22V

Vc(100ns) = 3.2V (98.5% not 98.2%)

And then using the 25ns voltage to get dV/dT

25ns --> dV/dT = (220V - 76.44V)/25ns = 5.7424E9 V/s

50ns --> dV/dT = (220V - 26.56V)/50ns = 3.8688E9 V/s

75ns --> dV/dT = (220V - 9.22V)/75ns = 2.8104E9 V/s

100ns --> dV/dT = (220V - 3.2V)/100ns = 2.168E9 V/s

ic(25ns) = 470pF x 5.7424E9 = 2.69A

I think you are right that at the instant the capacitor starts discharging, dV/dT will be large simply because no time has yet passed and dividing by zero gives big numbers.

But after just 1ns we'd get

Vc(1ns) = 220V x e^-(1ns/(470pF x 50.32ohm)) = 210.89V

dV/dT = (220V - 210.89V)/1ns = 9.11E9

ic(1ns) = 470pF x 9.11E9 = 4.28A

After you had warned me of the issues with high dV/dT affecting mosfets, I read that article I posted a few comments above. I can't remember exactly where, but they mentioned that the dV/dT is a problem for mosfets if this occurs after (not during) the switch has been triggered. I guess that is beside the point though because the dV/dT will still start off high once the gate is done switching and begins discharging the capacitor to the transducer.

Also, it was mentioned in the article that if the dV/dT is extremely large for even as little as 1 or 2ns, the mosfet could have a false trigger or damage the component permanently. If this is the case, how are these devices supposed to be used at all in an AC configuration with an frequency above a few kHz? (My intended use is not with AC, though)

I seem to be at the point where I'm on the edge of damaging the component which isn't good. If I do end up having to research another way to go about creating a "pulse width switch" that can interface with a microcontroller that would be a huge setback. It might be possible to avoid interfacing with a controller, and instead using a crystal timer that could control the switching, with the microcontroller telling the crystal when to begin sending pulses.

Anyways, I think you are misunderstanding that ic(t) equation. If the time constant is 25ns, the dV/dT wouldn't simply be 220/25ns, it would need to incorporate the voltage lost during that time.