I know that the maximum energy that can be stored in a capacitor is:
$$ U = \frac{Q^2}{2\,C}\,. $$
What I am asked is to say how much energy was stored when half of the electric charge that could be contained was deposited in the capacitor. I would naturally answer "a quarter", since energy depends on the square of the charge, but the correct answer should be "a half". Because? What am I doing wrong?
The correct answer would be a quarter. It is very much like the energy stored in a spring is
$$E = \frac{kx^2}{2}$$
Electrons repel each other. If you add electrons to a metal, the metal is charged. The electrons spread out, keeping as far away from each other as possible. It is a lot like little springs pushed them apart.
You can push another electron onto the metal, but it takes a force to do it.
The electrons get crowded a little closer together because there are more of them. The force between each one goes up.
If you push more electrons on, each one requires a little bigger force than the last.
