At What Temperature Do Copper and Iron Wires Have Equal Resistance?

[confused1]

A copper wire has a resistance of 0.501 ohms at 20.0 degrees C, and an iron wire has a resistance of 0.487 ohms at the same temperature. At what temperature are their resistances equal?

Resistivity of Copper: .0039
Resistivity of Iron: .0005

 

[Poncho]

Try this formula

$$\frac{{\Delta R}}{R_0} = \alpha \Delta T$$


You need to look up the alpha

 

[physbod]

For pure metals the relationship is (almost) linear and the formula is:
R=R₀[1+$$\alpha$$(T-T₀)

FYI the figures quote are temperature coefficients measured in K^-1 (degrees Celsius to the minus one is also fine, as the increments are the same. Just don't mix the two as values for T and T₀ in one calculation):

Where you want the resistance to be equal, you just back the copper and iron halves of the equation together (copper on the left, iron on the right).

0.501 [1 + 0.0039 (T - 20)] = 0.487 [1 + 0.0005 (T - 20)]

Multiply the lot out, so as to separate T
0.501 + 0.0019539T - 0.039078 = 0.487 + 0.0002435T - 0.00487

Real numbers on the left, expressions in T on the right
0.501 - 0.487 - 0.039078 + 0.00487 = (0.0002435 - 0.0019539)T
-0.020208 = -0.0017104T

Divide both sides by -0.0017104
T=11.81478017 degrees Celsius

It you subs in [T-20] as -8.185 into either equation to double check, at T, the resistance will be about 0.485$$\Omega$$

6 years late, but hope this helps.

 

[physbod]

On reflection, I have noticed that Iron has a resistance temperature coefficient of 0.005K^-1, not 0.0005K^-1. The principle of working through it is still correct, but this answer is only correct for the question asked incorrectly ...if that makes any sense

 

[rwisz]

The resistivity of a material is a measure of its resistance to the flow of electricity. It is affected by temperature, with most materials experiencing an increase in resistivity as temperature increases. This phenomenon is known as the temperature coefficient of resistivity.

In this case, we can see that the resistivity of copper is much higher than that of iron, meaning that copper has a higher resistance at the same temperature. This explains why the copper wire has a higher resistance than the iron wire at 20.0 degrees C.

To find the temperature at which their resistances are equal, we can use the equation for resistivity:

R = ρ * L/A

Where R is the resistance, ρ is the resistivity, L is the length of the wire, and A is the cross-sectional area.

Since we are comparing two wires of the same length and cross-sectional area, we can set up the following equation:

0.501 = (0.0039 * L)/A (for copper wire)
0.487 = (0.0005 * L)/A (for iron wire)

We can solve for L/A in both equations and set them equal to each other:

0.501/(0.0039 * L/A) = 0.487/(0.0005 * L/A)

Simplifying, we get:

128.46 = L/A

This means that at any temperature where the length of the copper wire is 128.46 times the cross-sectional area, its resistance will be equal to that of the iron wire.

Therefore, at a temperature where the length of the copper wire is 128.46 times the cross-sectional area, the resistance of both wires will be equal. This temperature can be calculated using the temperature coefficient of resistivity for both materials.