How Does Temperature Affect Electrical Resistance in Copper Wire?

[porporking]

Homework Statement

Hi Physicsforum, this is my first post so apology if I make unclear equation. I try to solve this question by pluging t0 as temp. at Valley and t as temp. at Antartitca directly as oppose to setting separate equation. I got different answer than the solution said

While taking photographs in Death Valley on a day
when the temperature is 58.0°C, Bill Hiker finds that
a certain voltage applied to a copper wire produces
a current of 1.00 A. Bill then travels to Antarctica
and applies the same voltage to the same wire. What
current does he register there if the temperature is
-88.0°C? Assume that no change occurs in the wire’s
shape and size.

The answer from solution book is 1.98 A

Homework Equations

ρ = [ρ₀(1+α(T-T₀))]

R = [R₀(1+α(T-T₀))]

The Attempt at a Solution

R_antart = [R_valley(1+α(T-T₀))]
V/I_antart = [V/I_valley(1+α(T-T₀))]

V are same so eliminate V on both side,

1/I_antart = [1/I_valley(1+α(T-T₀))]

I_antart = I_valley/(1+α(T-T₀))

substitute
I_antart= 1.00A/(1+(3.9*10^-3)(-88c-58c)) = 2.32 A

 

[mjc123]

Over such a wide temperature range, the equation ρ = [ρ₀(1+α(T-T₀))] is not a good approximation.
Start with the definition α = (1/ρ)dρ/dT and integrate from T₀ to T.

 

[gneill]

Homework Equations

ρ = [ρ₀(1+α(T-T₀))]

R = [R0(1+α(T-T0))]

The $T_o$ here is the reference temperature at which the value of $\alpha$ was determined. If you try to use some other temperature here (such as one of your sampled temperatures), then the equation will not describe the same line.

Luckily, the temperature coefficient you've quoted would appear to be the one for 0°C, so that your $T_o$ is actually zero.