As Robert Rapplean says above, the dangerous thing is the energy. To put it another way, you could take the jacketed slug out of a bullet, swallow it and likely pass it without issue. Bullets are not dangerous any more than the ground is, but when you fall off a building, the ground becomes lethal. It is the energy being dumped in to you quickly that is dangerous.
Here are some various energies with their use cases:
~500J: light target and handgun rounds (9mm, 45ACP, 40S&W)
often popular for submachineguns due to lower gas pressure making the recoil cycle easier to maintain reliably
~1500-2000J: lighter hunting/varmint rifles, carbines, modern battle rifles (5.56NATO, 7.62x39)
~2000-3500J: medium hunting/game rifles, previous century battle rifles, modern light machineguns, some modern heavier battle rifles (30-06, 30-30, .308, 7.62x51NATO)
greater than 3500J: rifles built for either large game, extreme range or are mounted/crew-served weapons
.. at this point your target is typically not people, but large animals
.. farther on, the target becomes vehicles, emplacements or areas to suppress
e.g. the M2 heavy machinegun, mostly unchanged for 100 years, fires around 13000J
Modern armor's job is to slow down a projectile. If you are hit, you are taking that energy. The difference between a a few broken ribs with a nasty bruise and a lethal injury is how fast you take that energy. Spreading that energy impulse out just a little is very significant. Armor penetrating rounds often defeat armor by going extremely fast, even if they are lower energy.
Momentum, often represented by 'p' is the product of mass and velocity:
p = m*v
In order to accelerate something up to that velocity, you apply force to it. When you stop the object, that stored force pushes back.
F = dp/dt
.. that is the force equals the change in momentum divided by the time for that change to take place.
If you plug in the definition for momentum, above:
F = (dvdm) / dt
.. let's presume mass is constant (if not, you have Star Trek acceleration)
F = (dvm) / dt
The force needed to achieve a certain change in speed over a certain time, or the force realized by a change in speed over a certain time, is the name of the game. So, can magnetic linear accelerators do this?
First, let's see how much force an M4 firing a M855A1 round imparts to the slug to bring it to lethal velocity:
Wikipedia lists the performance of the M855A1 FMJBT round as:
948m/s, 1859J, 4g
Wikipedia also lists the M4 carbine as firing the M855A1 FMJBT at:
Wikipedia lists the M4 barrel as:
I'll take the above as a baseline and round the values off a bit:
900 m/s, 1860J, 4g over 370mm
We're missing the time it takes to accelerate the bullet, but we can solve for that using the definition of a Joule:
1J = kg*m^2/s^2 =>
J / kg = (m/s)^2 =>
m / sqrt(J/kg) = s
0.0037 / sqrt(1860 / 0.004) = s
.. on to the force, now:
F = (dvm) / dt =>
F = 9000.004 / 0.00000543 =>
F = 662983
.. that's a fair bit of force.
The Wikipedia article on railguns is a bit of a mess. It has a useful amount of math, but it is smeared around over various aspects of the system. I'll try to trim it down a bit. Many railguns consist of two 'rails' that a sled or armature runs between (or on top of). The sled completes a circuit between the rails. The basic idea is that a moving current generates a magnetic field, which can apply force. A lot of current will get you a lot of force on the sled, which can push a projectile.
If current is "I" and the inductance per length of the rails is "L' ", the force applied to the sled is:
F = (L' * I^2) / 2
.. now it gets a tad hairy if you want to unravel that and actually compute stuff:
F = ( (mu0I^2) / 2pi ) * ln( (d-r) / r )
mu0 is a constant - it is the "vacuum permeability"... kinda the effect of generating a magnetic field using electricity in empty space with no physical constraints.
I is still current, or amps.
d is the distance between the exact centers of the rails, (disregarding their thickness - for now)
r is the radius of the rails (presuming they are circular cylinders)
The big contributer here is I, or current. It is in the numerator and is squared.
Let's shape the railgun to be roughly the same form factor as a M4.
.. let d be the width of the M855A1 round, or 5.56mm
.. let r be the width of the M855A1 round, divided by approx. e+1, or 1.495315mm (0.0014953143028)
there are arguments for thicker or thinner rails, and measurements that make sense, but honestly I chose this value to make the math easier.
Computing the parts we know and have decided:
F = ((0.00000126 * I^2) / 6.28) * ln( (0.00556-0.001495.. / 0.001495.. )
=> ((0.00000126 * I^2) / 6.28) * 1
F = 0.0000002 * I^2
So, to duplicate the force of an M4:
662983 = 0.0000002 * I^2
=> 1820691 = I
.. that's quite a fair amount of current.
Now, I'm doing nearly enough hand waving to take off, but that gives you a really rough ballpark measure.
I think that the picture I have is incomplete, because I know there have been railgun experiments done and I can't imagine them using that much power to get that mediocre of a result. That's what the math I was able to crib together says though, so that's what I'll go with.
So, to your actual question, how many MW do you need? Well, it depends on the number of volts. If 1 is enough, then about 2MW should be fine. I'd say hedge your bets and say at least 3.
Keep in mind that is for ONE shot, though.
If you want more penetration, that often means more energy. You can make your projectile lighter, but at some point you'll get to the point of firing a needle one molecule thick. If you want to propel a M855A1 round at M2 heavy machinegun energies or beyond, you need to scale up. It seems the force is limited by essentially the square of the current.
However! If you have this technology, there are a couple questions you need answers to, some limitations and secondary effects:
Issue: friction and strength
Railgun rails undergo a massive amount of friction. They need to be electrically conductive, and that limits the materials you can use to construct them. The M4 barrel is commonly made of a chromium-molybdenum steel alloy that handles high heat extremely well, but it is about as electrically conductive as a pineapple. In fact, the pineapple, due to the water and acids in it, likely conducts electricity better.
If you scale up the power enough to have a functional railgun, there are secondary effects that are usually pretty weak, but at those power levels are evident. Railgun rails experience a very strong shear force pushing away from the armature/sled. Your rails need to also be very strong or they'll simply bend away from each other, tearing away from the sled.
Solution? magic material
You mention you have superstrong armor. If your armor material is also electrically conductive, then it can be the rail material, too. Most ballistic armor works on the principle of slowing down the round rather than trying to deflect it. For instance, a metal shield will deflect a sword, but a sword can cut right through kevlar. Kevlar works by enmeshing a spinning ballistic round in a web of strong threads. The faster the bullet spins, the tighter it wraps itself up in the threads, slowing itself down.
Most materials get their 'strength' from molecular bonds. If you pick up a stick and try to snap it, the force you feel resisting you is bonds between the organic molecules that make up the stick. If your magic material was formed with sub-nuclear bonds, it might be much stronger.
Quantum entanglement is a process of getting two photons to exchange quarks. Once this occurs, measuring the spin of one photon causes the other to instantaneously (to our best measure) exhibit the opposite behavior. If a material is made that consists of molecules with some entangled particles, it might be incredibly strong. I don't know if anyone has tested, or if it is even possible, but there you are.
Effects: super strong and thin, no friction issues, conductive
If this material exists, it can be expensive and limited use, but it should have other effects on society. It should be possible to build bearings that effectively never wear out and are solid. If the entangled molecules can be aligned along one surface the material should be super smooth and nearly frictionless. Any vehicle should become way more fuel efficient since this material could be used to strengthen it with little weight. If it is also conductive, it could make electronics a lot more efficient, too. The additional links between molecules of the material could make it even easier for electrons to flow, reducing resistance and impedance. If you can move electricity faster, computers also get a lot faster. One of the main issues these days is getting a signal from one side of a chip to the other side. Additionally, one of the slowest things you can do in a computer have the CPU is talk to other hardware like network cards, drives, video cards, and the like. Faster signals mean all your devices in a computer get their data way faster, too.
If you have a material that is essentially superconducting at room temperature, it might make electricity flows so efficient that railgun power requirements go down significantly. *wink, wink*
If your material is very low resistance/impedance, that could solve the next issue.
Issue: power delivery and/or storage
Generating lots of power is one thing. Getting it from the generator to where it needs to go is another. A fundamental issue and a big loss of power is just moving it around. One of the issues with railguns is just switching in all the power and getting in to the rails all at once. As you saw, diameter of the rails is important. If you want to push a lot of power through a material, the thicker the material the 'harder' it is to get the power through, but the less you lose as heat. If you make the material thin, it is easier to move the power through, but if you make it too thin, you get a light bulb filament or a toaster heating element and you lose power as heat.
Solution? magic material
If your super material is not only strong but conductive, it could solve that issue. It could allow the switching of a vast amount of power without resisting the flow of the electricity, losing almost none as heat while also being so thin that the power is not impeded by the material.
Effects: most of the above, but even more so!
Electricity amounts required to do anything will plunge dramatically. Weaving some of this material in to wires would make delivery better. Wires of this material would carry power farther easier, meaning you need to put in less power at the front to get a certain amount at the back.
If you couple very low resistance/impedance highly conductive and strong material with enhanced power generation, power should not be a problem for anyone, anywhere. The amount of usable power from existing power generation methods would increase overnight. New electronics would require way less power, enhancing the effects of traditional generation sources. If you also have 2MW/s generators that are man-portable, then anyone can have almost unlimited power. Magnetic levitation should be widespread and common if the generators are as well. That much power easily available might also make directed energy weapons feasible. You might also hit the edge of the ability to synthesize matter at low scale. If you can put a 2MW generator in every soldier's weapon, having a few tens of Mtons per second of power and superconducting wire available might allow the creation of matter from energy. It wouldn't be fast, it would likely be pretty slow and painstaking, but it might be possible.