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Is this a correct application for the time dilation theorem?

Suppose an electronic clock has a copper wire of length $d$, which allows the electrons to take $t$ seconds to complete a cycle on the $d$-long copper wire, in the electronic clock.

Then, the the entire clock is moving in a rocket at a speed $v$ in a straight line.

The electronic clock must slowdown. My question is, is the following the new slowed-down time (update: in the reference frame of someone stationary that is not moving in the rocket)?

$$
t_{new} = frac{t}{sqrt{1-frac{v^2}{c^2}}}
$$
where $c$ is light speed.

Is that correct? If not correct, why?

Note: just to be clear, I know that applies on the light clock. But does it also apply on other non-light clocks, such as an electronic clock? E.g. should I replace $c$ by the speed of the electron movement on that copper instead?

To answer your first question, yes, the formula is correct, and it shows changed time interval for a moving observer. And, yes, the formula can be applied to all clocks and to all events.Time interval that takes one event to happen changes depending on the motion of the observer. In order to see this for all events, look for some derivation of Lorentz boosts that is, Lorentz transformations regarding observers in constant speed motion in regard to each other. One way to derive these is to observe what happens to a spherical light pulse for two different observers in relative motion. When you get Lorentz transformations for the time and space then you just check out how does a time interval behave, that is, how does difference betwen two time points look in one and how in some other frame. So you can have t1 and t2. Time interval is t1-t2...transform t1 and t2 and check how does transformed time interval, T1 - T2, behave.

Note that the clock slows down as viewed from someone else's reference frame.

In the rocket: clocks, melting ice, human biology, time, and whatever else you can think of all proceed normally. (Same with length contraction: right now we are both pancake flat in the reference frame of a cosmic ray, but we don't really notice it--at all).

Of course life on Earth proceeds normally too, even though the rocket's occupants see Earth's time ticking slowly (not to mention an, ahem, flat Earth).

All (properly functioning) clocks must experience the same time dilation, no matter what mechanism they use to tell time. Otherwise, you could use 2 different clocks to determine your absolute speed, but relativity says that absolute speed isn't a thing.

I recommend Feynmann's chapter on this. It is included in Six easy pieces I think. He points out that once you've found the effect for one type of clock (e.g. light pulse clock) then you have found it for all types, because otherwise the different types of clock would get out of step with one another when observed by one observer but stay in step when observed by another. That would be a direct impossibility. For example if one observer says that two clocks go "bong" at the same moment at the same place, then all observers must agree that the bongs happened at the same moment at the same place.

You can put your electronic clock next to a light pulse clock, both in the same state of overall motion, and then whenever they tick at the same moment in their joint rest frame, that is one joint 'bong'.

This is also discussed in other introductory texts of course.