Tungsten wires of all electric light bulbs

[discoverer02]

I'm looking for a jump start with this one. I'm having trouble getting started.

The tungsten wires of all electric light bulbs are designed to glow at about the same temperature. This requires, as a first approximation, that the power per unit surface area of the filament be the same for all.

a) Show that this leads to the requirement, at constant voltage, that r/l^2 is constant, where r is the radius and l is the length of the filament.

b) If P2/P1 = n is the ratio of the power consumption of two different light bulbs, show that r2/r1 = n^(2/3) and that l2/l1 = n^(1/3).

I've got lots of formulas, but I'm having trouble putting them together to show what a) and b) ask for. I'm sure once I get a), b) will follow easily.

P = I^2*R => I is current; R is resistance
R = pl/A => p is resistivity; A is surface area; l is length
A = [pi]r^2

P = I^2(pl/A) => P = I^2(pl/([pi]r^2))

P = V^2/R where V is potential
V^2 = (I^2)(R^2) => V = IR

So I'm going around in circles and getting nowhere.

Any clues would be greatly appreciated. I have a feeling the answer is staring me straight in the face, but I'm not seeing it.

Thanks much.

 

[Sonty]

Originally posted by discoverer02
that the power per unit surface area of the filament be the same for all.

I think this is talking about exterior surface so:
P₁/2πr₁l₁=P₂/2πr₂l₂
Then use P=U²/R=U²*π*r² / ρl

 

[discoverer02]

I need to learn to read the problem statement more carefully.

Thanks for your help Sonty.