Estimating Temperature and Wavelength of a Light Bulb Filament

[Brewer]

I have a question which is:

The filament of a 40W light bulb has a radius of 1.5μm and a length of 10cm.
Estimate:
i) the operating temperature of the filament
ii) the peak wavelength of the emitted thermal radiation

Stating any assumptions made.

I assumed that the filament was made of tungsten and has an emissivity of 0.26.

I then used the formula:
H = AeσT^4

but this gave me a temperature of 139972.5K, which as far as I'm aware is hotter than the surface temperature of the sun right?

For the second part of the question, I wanted to use E=mcΔθ to get the energy, and then use this with E=hf to find the frequency and thus be able to calculate the wavelength. However after further consideration I decided against this idea, as I don't have a change in temperature, only a single value (and the lack of specific heat capacity of tungsten in the question also lead me to think that this wasn't the best course of action to take).

Any help would be greatly appreciated.

 

[Tide]

I got about 7300 K for the temperature. Check your units and your math.

 

[Brewer]

Thanks for that. After a more indepth read of my textbook I found I had to use the surface area of the filament, whereas I had been using the cross sectional area of it.

I think the value that I got in the end was 7325.9K

And for the wavelength I got about 409.5nm. Do you agree with that?

 

[Tide]

That's close - I get about 396 nm.

 

[Andrew Mason]

I have a question which is:
The filament of a 40W light bulb has a radius of 1.5μm and a length of 10cm.
Estimate:
i) the operating temperature of the filament
ii) the peak wavelength of the emitted thermal radiation
Stating any assumptions made.
I assumed that the filament was made of tungsten and has an emissivity of 0.26.
I then used the formula:
H = AeσT^4
but this gave me a temperature of 139972.5K, which as far as I'm aware is hotter than the surface temperature of the sun right?
For the second part of the question, I wanted to use E=mcΔθ to get the energy, and then use this with E=hf to find the frequency and thus be able to calculate the wavelength. However after further consideration I decided against this idea, as I don't have a change in temperature, only a single value (and the lack of specific heat capacity of tungsten in the question also lead me to think that this wasn't the best course of action to take).
Any help would be greatly appreciated.

I think the radius of the wire is wrong. It cannot be $\mu m$. That is way too thin. It might be 1.5 millimeters but that seems a little thick for a light filament. If you use 1.5 mm, the answer is about 1600 deg. K which is in the right order of magnitude.

AM

 

[Brewer]

I agree it seems far too thin, but I'm working off the question I have infront of me. Its either 1.5 $\mu m$ or 15. There was a mark on the page just where you'd put a decimal point so whether its a decimal point or a photocopying error I don't know