The speed of light, frequently referred to as “c” in lower case typically is used to describe what happens to light speed in vacuum which is the speed at which it can be able to go. It’s around 300,000,000 meters per second.
The initiation of electric field or electrical signalling by electrical signals within the wire is more slow. It could range between 50% the c, to as high as 99percent of c according to the wire’s insulation composition and. Read the Wikipedia page Velocity ratio.
There’s a second speed, which is called the drift velocity. It’s like following an individual electron, and it’s extremely slow in the range of the fraction of meters per second. It is difficult to visualize the propagation of electrical signals as one electron or as a bunch of electrons moving across this wire, from one side to another at 75% what light travels at… it is important to be aware that electrons bump into one another and exchange places using the help of a few holes that propagate the field. It’s similar to hearing sounds from a distance that is 1,000 feet. The sound that you hear in your ear isn’t the air molecules that were originally moving 1,000 feet away, but rather the compression wave that moves from molecules to molecules over the distance. See wiki article Drift velocity
But, when determining the time until the light turns on, two aspects are likely to have a greater significance more than speeding of the electricity through the wires:
- The mechanical motion of the switch can take several miliseconds before it is able to stop the contact.
- Phase angle is the angle at which you receive AC power is determined when you flip the switch. If the Sine is in the zero, there could be a delay up to 5 milliseconds before reaching the maximum voltage.
- The rate of heating of the filament is quite high; I would anticipate to take a few milliseconds before it reaches full brightness because it is heating the filament’s thermal mass.
- A couple of feet of wire is approximately a nanosecond per foot of travel delay. one nanosecond is approximately 1 millionth of milliseconds.