- #1

- 10,544

- 2,323

## Homework Statement

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..

when it comes to the filament in a lamp the book said the thinner part of the filament is hotter than other parts and evaporation occurs faster there because of smaller cross sectional area and higher resistance. This makes the thinner part even thinner until it breaks.

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

## Homework Equations

Ohms law

## The Attempt at a Solution

[/B]

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..

V

_{R2}= V * R

_{2}/(R

_{1}+R

_{2})

The power dissipated in R

_{2}is

W

_{R2}= (V * R

_{2}/(R

_{1}+R

_{2})

^{2}/R

_{2}

If you expand and simplify you get..

W

_{R2}= (V

^{2}* R

_{2}) / (R

_{1}

^{2}+ 2R

_{1}R

_{2}+ R

_{2}

^{2})

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?