Relationship between changing a wire's diameter and change in temp

[alanajames]

Homework Statement

Hello! I have a problem where I need to develop and justify a hypothesis surrounding the relationship between the thickness (or diameter) of a nichrome wire and the temperature of water that it's submersed in over a time of 5 minutes. (proportionality statement and equations)

Homework Equations

So far, I know that resistance is inversely proportional to cross-sectional area, and therefore:
R ∝ 1/A
R ∝ 1/∏r^2
R ∝ 1/∏(d/2)^2
and because 2 and ∏ are constant
R ∝ 1/d^2 (resistance is inversely proportional to diameter squared) (I think!)

Now, I've also researched Joule's law, which eventuates into:
Heat produced = Pt = VIt = V^2/Rt = I^2Rt

The Attempt at a Solution

Now, I'm trying to get a relationship between diameter and ΔT, and I thought, so I have R ∝ 1/d^2, and P = V^2/R where V is a constant so P ∝ 1/R
and since P is the change in heat:
ΔT ∝ 1/RORsince thickness makes resistance go down
and less resistance makes ΔT (somehow)
then is R ∝ ΔT
and substituting so therefore:
ΔT ∝ 1/d

and I don't have much idea how to reason out what the graph will look like/why...

I think I've tied myself in knots... any help would be much appreciated! Thank you!:)

 

[CWatters]

I think you have to assume the wire is connected to a constant voltage source and I'd also ignore any change in resistance of the wire with temperature.

Therefore your approach is correct..

R ∝ 1/∏(d/2)²
R ∝ 1/d²

Then use..

Power = V²/R

Power = V² * d²........(1)

If the water is in an insulated container then the energy it contains (E) is given by

E = Specific heat capacity * mass * Δ Temperature

Rearrange to give..

Δ Temperature = E / Specific heat capacity * mass

Then..

E = Power * Time

so

Δ Temperature = Power * Time / Specific heat capacity * mass ...(2)

Put (1) into (2)...

Δ Temperature = V² * d² * Time / Specific heat capacity * mass

Voltage, Time, Specific heat capacity and mass are all constant so

Δ Temperature ∝ d²

eg The change in temperature over 5min (not the actual temperature) is proportional to d²

Temperature = Initial Temperature + Δ Temperature

So actual temperature is proportional (but not directly proportional) to d².

http://en.wikipedia.org/wiki/Proportionality_(mathematics )