Resistance of a light bulb changing with temperature

[Khuwwa]

Homework Statement

The resistance of the tungsten filament in a light bulb changes with temperature according to equation (1), below.
I am supposed to calculate the temperature of the light bulb at voltage 0.8280 volts and a current of 0.0853 amps. The bulb is connected to a 1.5915 volt battery. The bulb's resistance is 2.28 ohms.

Homework Equations

R = Ro(1+a(T-To)).........(1)

T = (-1a^-1)+(a^-1)(Ro^-1)(R)+T...(2)

Where Ro is the resistance at the initial temperature To, and R is the resistance at the final temperature, T. a is the coefficient of resistivity.

The Attempt at a Solution

The coefficient of resistivity for tungsten is 0.0045 degrees Celsius ^-1.
I assumed To to be 20 degrees Celsius.
I assumed Ro to be 2.28 ohms, the resistance of the bulb alone.
I want to find T, but I didn't have R yet, so I calculated it with:

R = voltage across bulb (V) / current through bulb (A)
= 0.8280 V / 0.0853 A
= 9.71 ohms

I put my values for Ro, R, To, and a into equation (1). I got T = 20, the same as To, so I checked two different online equation solvers. They told me I had made a mistake in the calculation, and that T was actually 744 degrees Celsius. I checked this on equation (2) and got the same answer. I knew this couldn't be right, so I switched my values for R and Ro. The solver and equation (2) both had T = -147 degrees. This does not seem right.

My question is, where did I go wrong? I am sure that To and a are correct. I suspect I have the wrong R value. How can I find the correct value for R? Is Ro correct? Any help will be greatly appreciated. This is due on April 2, and it has me stumped.

 

[berkeman]

Equation #2 doesn't look correct to me. a by itself isn't useful -- seems like it needs to be more in the form of 1+a, like you have in the first equation. Plus the units don't seem to match up for each term in that equation. Each term should have units of degrees C (or degrees Kelvin?), but on the RHS the first two terms look to have units of 1/degrees C.

And just to check (I don't know offhand) -- you're supposed to use degrees C or absolute temperature in Kelvin?

 

[marcusl]

Homework Statement

The resistance of the tungsten filament in a light bulb changes with temperature according to equation (1), below.
I am supposed to calculate the temperature of the light bulb at voltage 0.8280 volts and a current of 0.0853 amps. The bulb is connected to a 1.5915 volt battery. The bulb's resistance is 2.28 ohms.

Homework Equations

R = Ro(1+a(T-To)).........(1)

T = (-1a^-1)+(a^-1)(Ro^-1)(R)+T...(2)

Where Ro is the resistance at the initial temperature To, and R is the resistance at the final temperature, T. a is the coefficient of resistivity.

The Attempt at a Solution

The coefficient of resistivity for tungsten is 0.0045 degrees Celsius ^-1.
I assumed To to be 20 degrees Celsius.
I assumed Ro to be 2.28 ohms, the resistance of the bulb alone.
I want to find T, but I didn't have R yet, so I calculated it with:

R = voltage across bulb (V) / current through bulb (A)
= 0.8280 V / 0.0853 A
= 9.71 ohms

I put my values for Ro, R, To, and a into equation (1). I got T = 20, the same as To, so I checked two different online equation solvers. They told me I had made a mistake in the calculation, and that T was actually 744 degrees Celsius. I checked this on equation (2) and got the same answer. I knew this couldn't be right, so I switched my values for R and Ro. The solver and equation (2) both had T = -147 degrees. This does not seem right.

My question is, where did I go wrong? I am sure that To and a are correct. I suspect I have the wrong R value. How can I find the correct value for R? Is Ro correct? Any help will be greatly appreciated. This is due on April 2, and it has me stumped.

Your equation (1) gives the correct answer, 744 degrees. What did you do to get a different answer?

I have no clue what eq. (2) is...

 

[Khuwwa]

Berkeman, a was given to me in the assignment, and that is how I am sure it is correct. I actually checked the units for equation (2), because I wasn't sure, and I got degrees Celsius... but I just tried again and I'm not sure! I'm going to try the calculation again, thank-you for pointing that out...
marcusl, I am just no good at algebra! The equation is correct, it is R that I am not sure about. Equation (2) is just a re-arrangement of equation (1).

 

[Khuwwa]

I tried equation (2) again, this time paying more attention to the units. The answer comes out as 744 degrees Celsius, so it's not the wrong equation, but again, my R value.

I found something in my notes saying Ro is correct, so it is R for sure.

I need T in units of degrees Celsius.

 

[Khuwwa]

R must be wrong, because all the other variables are correct, but the temperature can't be 744 degrees! I just want to know what value I should use for R.

 

[marcusl]

What's wrong with 744 degrees? Should it be higher?

 

[Khuwwa]

I think the temperature wouldn't rise that much from 20 degrees. 744 degrees is really hot, shouldn't it be more like 25 degrees or something?

 

[Khuwwa]

I'm supposed to make a graph of temperature vs. voltage. I actually have a number of voltages at which I'm supposed to calculate the temperature.
Thank-you for helping, I was afraid my thread would be forgotten.

 

[marcusl]

The color emitted by a "black body" depends on temperature. 744C would glow a dull red like an electric range element. A 100W light bulb operates at around 2400K or 2100C, so the filament is made of tungsten which won't melt until around 3700K.

 

[Khuwwa]

This was one of those really small lab light bulbs, I don't know the wattage...
Do you think I haven't made a mistake?

 

[Khuwwa]

Thanks for the help!