Lamp filament

arcnets

Hi all,
I have two light bulbs, both for 230 volts. One has 100 watts, the other 15 watts.
It is clear that the 15W bulb has the higher resistivity. But is its filament
a) longer
b) thinner
c) both,
and why?

Thanks!

Chi Meson

Clear up this small detail: resistivity is not resistnace. Resistaivity is a property of the material that the filament is made of. The length and cross-sectional area of the wire, along with the resistivity of the wire, determine the resistance of the wire.

So your logic so far is correct, but it is clear that the 15 W bulb has the largest resistance. All of the bulbs' filaments have (I assume) the same resistivity.

Chen

Resistance is given by:

[tex]R = \frac{\rho l}{A}[/tex]

Therefore:

[tex]R\ \alpha\ \frac{1}{A}[/tex]

[tex]R\ \alpha\ l[/tex]

So longer filament means larger resistance, and thinner cross section also means larger resistance. [:)]

arcnets

Thanks. But how about the heating of the wire, and how about radiation? That must also play a role, mustn't it?

Chen

The temperature of the wire does affect its resistance, if I'm not mistaken it's:
[tex]R_T = R_0(1 + \alpha T)[/tex]
Where [itex]\alpha[/itex] is a property of the material the wire is made of (similar to the resistivity [itex]\rho[/itex]).

But I don't see how this would affect the answer? It's true that the resistance of the wire is not constant, but the numbers specified on the bulbs refers to their power at the same temperature (might be room temperature, but doesn't really matter actually). So at the same temperature, the resistance of the 15W bulb is higher than that of the 100W bulb and the answer remains.

However we can't know for certain if the wire is thinner, longer or both. It could be really really long but just as thin, or it could be really really thin but just as long, or it's both thinner and longer.

arcnets

That's exactly my question. Well, you need a certain surface area so the wire can radiate at the correct power...

Chen

Ok, surface area depends both on the length and area of the wire and is proportional to both of them. So longer wire means more resistance but also more radiation, but we know that the bulb with the higher resistance emits less light.

Still, though, the wire can either be:
a) somewhat thinner
b) a lot more thin but also a bit longer
There is probably some fixed ratio between how much thinner and how much longer the wire should be.

Chi Meson

If the wire is too long, then it will not get to the correct temperature to radiate visible light. It still could radiate at the same power, but it would seem even dimmer to us since more of its light would be infra red. You could make the filament really thin, and have a "whiter" light but at a low intensity. No matter what the power of the bulb, the "color" of the light depends on the temperature of the filament.

arcnets

Thanks - I bought 2 bulbs and it looks like the 100W has the longer and wider filament. It is coiled, too, while the 15W is not coiled.