How exactly does a bulb light up? 
A typical value for the electron drift velocity in a copper wire is $10^3\ \mathrm{m\ s^{-1}}$. In the circuit below, the length of the copper wire joining the negative terminal of the batter to the lamp is $0.50\ \mathrm{m}$.

(i) The switch S is closed. Calculate the time it would take for an electron to move from the negative terminal of the battery to the lamp.
(ii) The lamp lights in a time much less than that calculated in (e)(i). Explain this observation.

In the second part, I can't imagine the situation. I reckon that not all electrons travel with a drift velocity, so the faster ones reach the bulb and make it glow.
But how exactly does this lighting happen? The electron comes in contact with the circuitry and lights up the bulb, or is it because of the electric field?
 A: A single electron takes some time to move from the battery to the bulb but the lamp lights up faster than that. The reason is due to the fact that it is not the electrons travelling from the battery that light up the lamp when it is first lit, rather due to nearby electrons. The electric field is set up almost instantaneously in the circuit due to movement of electrons from their initial position all over the wire,(at the speed of nearly c, depending on the medium) and the electrons nearer to the lamp pass through the circuit lighting it. So even if the drift velocity of the electrons is so small, the lamp gets lit up.
A: We must understand that the internal electromagnetic forces inside the wire constrains the charge movement.
It is not so different from a rigid body as a rocket or a ship. The source of power is at the rear in both cases, but the rear part can move only if all body moves. A ship can have a slow velocity, but as soon as its rear part starts to move, its front part (say 100 m ahead) also moves.
It is not possible for individual electrons to flow through the wire unless some of its fellows ahead also moves. Otherwise an excess of negative charge would be built in regions of the wire.
A: Imagine the analogy of water flowing in pipes. You have a pump which pushes water into one pipe and sucks it out of another. There is a valve which stops the flow of water. And there is a turbine with fans which are connected to a friction point, all the force of the turning turbine goes to heat a little bit of metal white-hot to make a lamp.
The pump is regulated to produce just so much pressure difference between the intake and the outtake. So when there is no water flowing, it just keeps that pressure at its constant voltage. Then the valve opens. The water can move as fast as the turbine lets it, which is very slow becaue the turbine has a lot of resistance. There is still a voltage (pressure) drop across the turbine.
It takes a little while for the light to light, because the turbine has to spin fast enough to heat that filament white-hot. Also, the moving water has inertia, and it takes time for it to build up speed. (Hysteresis.) But it doesn't matter how long it takes for water to get from the pump to the turbine, as long as the pressure stays at the right level at the turbine.
It isn't the amount of flowing water that gives power to the turbine, it's the pressure.
Also, the amount of water pressure can be exactly measured by the amount of pressure on the walls of the pipe. So we can say that there is in fact no pressure inside the water, there's only pressure on the pipe walls and the turbine. I'm not sure why we'd want to say that, but we could. We'll calculate the right answers if we say that.
Is this analogy useful? Maybe think about it before you decide....
