> Also, the energy of an electromagnetic wave grows as the 4th power of its frequently, IIRC. This is why old power supplies which directly used the 60Hz frequency from the socket were so bulky, while modern power supplies, which convert the mains electricity to high-frequency AC, can use physically small and lightweight transformers.
It's much more roundabout than that.
A transformer has a magnetizing inductance (you can measure this with an LCR meter by leaving all other windings open). This is generally a pretty big inductance (several henries for mains transformers). When you supply a waveform, that inductance will cause a small current to flow. That's the magnetizing current. The problem is saturation - if the voltage over time gets too large [=the magnetic field in the core], the transformer's core will saturate (that's also why transformers cannot handle DC; any DC offset will accumulate over time and cause saturation). This causes the magnetizing inductance to drop to approx. zero. When that happens, the magnetizing current becomes huge, basically a short circuit.
This means that you need to avoid saturation. Note that the strength of the field depends on the number of turns - each turn reduces the impressed field. So you want as many turns to avoid saturation as necessary, but as few turns as you can get away with (each turn adds resistance, and needs space inside the transformer).
And this is why mains transformers are so big. Because the frequency is low, voltagetime is large. So they need many, many turns (a few hundred to thousands) in order to prevent their steel cores from saturating. So the primary winding is a very long wire, which has a lot of resistance. So you need to increase the size of the core to fit a thicker wire, in order to avoid resistive losses. Now that bunch of wire doesn't fit any more through your transformer, so you need to use a bigger core, but wrapping N turns around that needs even more wire and so on. So you end up with a big, heavy transformer.
For a high-frequency switch mode power supply, the frequency is high, so voltagetime is orders of magnitude lower (e.g. 100 kHz vs. 50 Hz = 2000x lower), so they need far fewer turns. So their primary winding's wire is short, so it can be a) a little bit thinner and b) takes up much less space! So you get a small, light transformer.
But the core, it doesn't really care about power, like at all. It's all about not driving it into saturation, and having windings that can carry your current. Those two factors make the cores of mains transformers so big.