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 =0.00480 (C°)-1 and a resistivity of 4.90 x 10-5Ω m. For the second, the temperature coefficient is α2 = -0.000600 (C°) -1 and the resistivity is 4.60 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? Ignore any changes in length due to thermal expansion.
To solve this problem, we need to relate the resistivity, temperature coefficient of resistivity, and length of the wires.
Let's denote:
- Resistivity of the first wire as ρ1 = 4.90 x 10^-5 Ω m
- Resistivity of the second wire as ρ2 = 4.60 x 10^-5 Ω m
- Temperature coefficient of resistivity of the first wire as α1 = 0.00480 (C°)^-1
- Temperature coefficient of resistivity of the second wire as α2 = -0.000600 (C°)^-1
- Length of the first wire as L1
- Length of the second wire as L2
We are given that when the wires are joined, the total resistance does not change with temperature. From this information, we can conclude that the resistances of the two sections are equal at all temperatures.
The resistance of a wire can be calculated using the formula:
R = ρ * (L / A)
Where:
R is the resistance
ρ is the resistivity
L is the length
A is the cross-sectional area
Since the cross-sectional areas of the two wires are the same, we can write the equation for the resistances of the two sections as:
R1 = ρ1 * (L1 / A)
R2 = ρ2 * (L2 / A)
Since the total resistance does not change with temperature, we can set these two resistances equal to each other:
R1 = R2
Using the formulas for resistances and rearranging the equation, we get:
ρ1 * (L1 / A) = ρ2 * (L2 / A)
Canceling out the common factor of cross-sectional area A, we have:
ρ1 * L1 = ρ2 * L2
Now, substituting the values we know:
(4.90 x 10^-5) * L1 = (4.60 x 10^-5) * L2
Rearranging the equation to solve for the ratio of lengths, we get:
L1 / L2 = (4.60 x 10^-5) / (4.90 x 10^-5)
Calculating this ratio, we find:
L1 / L2 = 0.9388
Therefore, the ratio of the length of the first section to the length of the second section is approximately 0.9388.