Temperature Coefficient of Resistance of Two Wires in Series

1. Apr 22, 2014

chap126

1. The problem statement, all variables and given/known data
Two wires A and B are connected in series at 0oC and resistance of B is 3.5 times that of A. The resistance temperature coefficient of A is 0.04% and that of the combination is 0.1%. Find the resistance temperature coefficient of B.

2. Relevant equations
Rt=R0(1+α0t)
αt0/1+α0t

3. The attempt at a solution
This is an example problem form A Textbook of Electrical Technology Vol. I. So the solution comes from that book:
It is seen that RB/RA = 3.5 = 0.003/(0.001-α) --> α = 0.000143oC-1

My problem is that this relationship is not clear to me. From the looks of the given solution it is saying that:
RB/RAAABABB and then solving for αB. I'm confused if this is the correct relationship and if so where it came from? How do you relate them without at least one other measurement at some other temperature?

Last edited: Apr 22, 2014
2. Apr 22, 2014

chap126

Ok, I think I figured it out. The given solution was way too simplified, and I'm still not sure how the author worked it out, but here's what I got:

From the given equation we know

Rt=R0(1+α0t) --> R1,A=R0,A(1+α0,A*1)

and because A and B are in series we also know that

-->R1,A+R1,B=(R0,A+R0,B)(1+α0,AB)

since the problem tells us that R0,B = 3.5R0,A, we can solve for R1,B in terms of R0,A

R1,B=(R0,A+3.5R0,A)(1+α0,AB)-R0,A(1+α0,A)

we also know that R1,B=R0,B(1+α0,B)= 3.5R0,A(1+α0,B)

Then set the two equations equal and solve for α0,B as the problem asks

3.5R0,A(1+α0,B) = (R0,A+3.5R0,A)(1+α0,AB)-R0,A(1+α0,A)

α0,B=((1+3.5)(1+α0,AB)-(1+α0,A)-3.5)/3.5
α0,B= 1.42x10-4

If anyone can explain how the author solved it, I'd be happy to see it. His way seemed much faster.

3. Apr 23, 2014

rude man

The given answer can't be right. αB, the coefficient of the higher resistor, must be greater than the coefficient of the entire resistance α since αA, the coefficient of the smaller resistor is less than α. But your teacher "sees" α = 0.000143 oC-1 which is < 0.001 = α. In fact, he/she has αB < αA!

Do it like this: assume R = T = 1 without loss of generality.
Then 1(1+αA) + 3.5(1+αB) = 4.5(1+α).

4. Apr 23, 2014

chap126

Now I'm really confused. The answer I got matches the one in the book, but this way gives a different answer. Where did I go wrong? Any hints?

Last edited: Apr 23, 2014
5. Apr 23, 2014

rude man

Problem is, it's often not easy to work through posters' math when a lot of it is posted.

I think my equation is straight-forward enough that it should be understandable. It's just

R(1+αAT) + 3.5R(1+αBT) = (R+3.5R)(1+αT)

with R = 1 ohm and T = 1 deg. C.

It should also be intuitively obvious that αB has to be > α since αA < α.