# Calculating the temperature of a conductor from the current through it

#### alexchamp29

I'm not sure if this is the right place for this but I have a question about the relationship between current through a conductor and the heat dissipated by the material.
Given the current, resistance, and specific heat of a material as well as the specific spacial dimensions is there a way to calculate the temperature of the material?
Obviously the temperature would change over time as heat is dissipated and resistance changes and I would imagine different parts of the material would heat at different rates.
I'm thinking about a simple copper wire with a current running through it. Maybe one if you guys can help me out with this, thanks.

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#### mitochan

Hello,
Given the current, resistance we can calculate generated heat energy and given specific heat of a material as well as the specific spacial dimensions we can calculate temperature raised by produced heat energy. Additional important information we need is leak of heat energy to environment by conduction, air circulation, etc. that should be incorporated in boundary conditions of heat conduction equation. Please be aware that I said just according to principles of physics. Engineering and technology point of view should be necessary for practical use.

#### DEvens

Gold Member
Probably the easiest way is a set of careful measurements.

You could go to a lot of trouble with thermal-hydraulic calculations to estimate the heat escaping to the surrounding area as a function of the heat flow in the wire. Note that the resistance of the wire is going to depend on the temperature of the wire. So that's going to be a nice bit of feedback. Also, you would need to know something about the surrounding conditions and the heat transfer from wire to those conditions.

Another approach might be to measure the resistance at various temperatures. That would allow you to build a correlation that says, if the resistance is this much then the temperature must be so much. So you apply a voltage and measure the current, then calculate the resistance. Then you can estimate the temperature.

#### alexchamp29

Ok I like the idea of measuring resistance and forming a relationship to temperature.
The procedure I have in mind involves heating a wire to between 100 and 500C. I don't think an IR thermometer would be an accurate way to monitor the temperature and I feel like a thermoresistor also wouldn't work. That's why I was asking if there was a way to crunch the numbers based on current and resistance.

#### Baluncore

The resistance of a pure metal like copper is proportional to absolute temperature.
What are the dimensions of the wire, and will you heat it with AC or DC current.
What are you trying to do with the hot wire.
There may be a better solution if we knew the application.

#### alexchamp29

Basically a vaporizer. Where a coil heats a liquid to a specific temperature for a specific amount of time. I just can't think of a good way to control the temperature with a thermoresistor so I figured there must be a quantifiable relationship between current, resistance, and specific heat.

#### BvU

Homework Helper
Given the current, resistance, and specific heat of a material as well as the specific spacial dimensions is there a way to calculate the temperature of the material?
With $P = I^2R$ you get the power dissipated and converted into heat.
Power is energy (heat) per unit of time , or $P = {dQ\over dt}$

Heat warms up the wire and the heat needed to warm up the wire is $Q = m\, c_p\Delta T$.

In other words, if $m$ and $c_p$ are constant, $P$ gives you the rate of warming up ${dT\over dt}$.

This only holds for a short time, as the wire will start giving off heat to its environment when its temperature increases. That part is the difficult part in your scenario: the rate at which heat is given off is difficult to estimate.

Has this become clear to you from the replies you have gotten so far, @alexchamp29 ?

#### Baluncore

Basically a vaporizer. Where a coil heats a liquid to a specific temperature for a specific amount of time.
If you know the mass and properties of the liquid, you can deliver sufficient energy to vaporise the liquid. The evaporating liquid will cool the heating element until all is evaporated, when the temperature of the heating element will rise suddenly.

We still have no idea of the scale. Is it one drop, or one litre of liquid?
Is it a continuous flow or an intermittent pulse of liquid?

If the thermal capacity of the heating element was significantly greater than needed to evaporate the liquid then the response time of the temperature controller would not need to be particularly fast.

#### BvU

Homework Helper
'Vaporizer' or evaporator ?

#### alexchamp29

So if I wanted a particular change in temperature I would multiply that difference by the mass and specific heat of the material and that would give me the power need to cause that change in temperature?

#### Baluncore

Yes.
But if there will be a phase change, additional energy will be required to rise through the temperature of the liquid to gas transition. The specific heat will differ for the liquid and gas phases.

"Calculating the temperature of a conductor from the current through it"

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