1. The problem statement, all variables and given/known data

Not homework but this seems the best place to post this. In a private conversation someone posted this explanation from a book on how a lamp filament fails..

I'm having some trouble doing the maths to show that the thinner higher resistance part gets hotter...

2. Relevant equations
Ohms law

3. The attempt at a solution

Imagine the filament is made up of two resistors in series R_{1} and R_{2} (See diagram). Let R_{2} be the part that's getting thinner for some reason...

The voltage on R_{2} is given by the potential divider rule..

So if R_{2} increases the power in R_{2} decreases due to the R_{2}^{2}. That should reduce the temperature in R2 which is the opposite of what the book suggests. Have I made an error?

I suspect the problem is to do with the physics of the filament. eg the thinner part has less surface area so although the power dissipated in that bit is lower the surface area is also lower so it's harder to dissipate the power.

If R2 is small compared with the total resistance, the current through the combination will be roughly constant. The dissipation in in R2 will be I^2 x R2, so it is proportional to the resistance of R2. In your formula, R2 appears in both top and bottom, so your statement is not quite accurate.

If R2 is the part getting thinner then, assuming R2 is same length as the rest of the filament R1, R2 will be bigger than R1, and
power in R2 = i*R2 = [V/(R1 + R2)]R2 which is > [V/(R1 + R2)]R1.