Two wires have the same cross-sectional area and are joined end to end to form a single wire. The first wire has a temperature coefficient of resistivity of α1 =0.00603 (C°)-1 and a resistivity of 5.00 x 10-7Ω m. For the second, the temperature coefficient is α2 = -0.000670 (C°) -1 and the resistivity is 6.20 x 10-5Ω m, respectively. The total resistance of the composite wire is the sum of the resistances of the pieces. The total resistance of the composite does not change with temperature. What is the ratio of the length of the first section to the length of the second section?
(1/α)=R*(L/A), solve for A
α*R*L=A
1/α1 =1/0.00603 (C°)-1=5.00 x 10-7Ω m *(L/A)
Solving for A,
(0.00603 (C°))*(5.00 x 10-7Ω m) *(L1)=A
Solving for A,
1/α2 = 1/-0.000670 (C°) -1= 6.20 x 10-5Ω m *(L/A)
(0.000670 (C°) -)*(6.20 x 10-5Ω m)*(L2)=A
Set equations = to each other since both areas are the same.
α1*R1*L1=α2*R2*L2
(0.00603 (C°))*(5.00 x 10-7Ω m) *(L1)=(0.000670 (C°) -)*(6.20 x 10-5Ω m)*(L2)
L1/L2=α2*R2/α1*R1
[(L1)/(L2)]=[(0.000670 (C°) -)*(6.20 x 10-5Ω m)/(0.00603 (C°))*(5.00 x 10-7Ω m)]
Plug in your values and solve. Verify for yourself or wait for someone else to verify on this post.
To solve this problem, we need to consider the relationship between resistance and resistivity.
The resistance (R) of a wire is given by the formula:
R = ρ * (L/A),
where ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area.
In this case, since the two wires have the same cross-sectional area, we can simplify the formula to:
R = ρ * L.
Let's assume that the length of the first section is x and the length of the second section is y, where x and y are in the same units (e.g. meters).
The total resistance of the composite wire is the sum of the resistances of the individual sections:
R_total = R1 + R2,
where R1 is the resistance of the first section and R2 is the resistance of the second section.
Using the formulas above, we can write:
R_total = ρ1 * x + ρ2 * y.
According to the given information, the total resistance of the composite wire does not change with temperature. This means that the resistances of the individual sections must cancel each other out, i.e., R1 must be equal to -R2.
Therefore, we have:
ρ1 * x = -ρ2 * y.
Now, let's substitute the given values:
(5.00 x 10^-7) * x = -(6.20 x 10^-5) * y.
Dividing both sides of the equation by (6.20 x 10^-5), we get:
(5.00 x 10^-7) * x / (6.20 x 10^-5) = -y.
Simplifying further:
(5.00 x 10^-7) / (6.20 x 10^-5) = -y / x.
Finally, calculating the value on the left-hand side of the equation:
(5.00 / 6.20) x (10^-7 / 10^-5) = -y / x.
0.806 x 10^-2 = -y / x.
Therefore, the ratio of the length of the first section (x) to the length of the second section (y) is approximately 0.806 x 10^-2.