Why is the exponent on the Stefan-Boltzmann law 2.78 in my experiment?

[ideasrule]

Homework Statement

A few weeks ago I did an experiment that involved empirically determining the exponent on the Stefan-Boltzmann law. I used a light bulb and measured the voltage and current across it for different voltages. Since P=k*Tⁿ and R=cT (i.e. resistance of the filament is proportional to temperature), P=k*(R/c)ⁿ. Taking the logarithm of both sides gives ln P=const + nlnR. At equilibrium, the bulb should emit just as much power in the form of blackbody radiation as the power supply provides, so P=VI.

I graphed ln P vs. ln R and measured the slope of the line: 2.78, a far cry from 4. However, the line was nearly perfect! All of the points were nearly touching the line of best fit I drew! Moreover, many other people did this experiment, and almost all of them got 2.5-2.9.

My question is: why 2.78?

The Attempt at a Solution

I'm thinking that if R doesn't increase linearly with T but is instead proportional to a power of T, the value of 2.78 would make sense. However, that's clearly an ad hoc approach. I've no idea why R would be proportional to anything other than T.

Edit: Please reply quickly, because I'm kind of on a deadline.

 

[ehild]

My question is: why 2.78?

The Attempt at a Solution

I'm thinking that if R doesn't increase linearly with T but is instead proportional to a power of T, the value of 2.78 would make sense. However, that's clearly an ad hoc approach. I've no idea why R would be proportional to anything other than T.

Edit: Please reply quickly, because I'm kind of on a deadline.

You are right, the temperature dependence of the resistivity of the tungsten wire in the bulb is rather a power 2 relationship at high temperatures than linear.

http://hypertextbook.com/facts/2004/DeannaStewart.shtml

ehild