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? Ignore any changes in length due to thermal expansion.
To solve this problem, we need to consider the relationship between resistance, resistivity, length, and cross-sectional area of a wire.
The resistance of a wire can be calculated using the formula:
R = ρ * (L/A)
Where:
R is the resistance of the wire,
ρ (rho) is the resistivity of the material,
L is the length of the wire,
A is the cross-sectional area of the wire.
In the given scenario, we have two wires joined end to end to form a composite wire. We can calculate the resistance of each section of the composite wire separately, and then add them to find the total resistance.
Let's assume the length of the first section of wire is L1 and the length of the second section of wire is L2.
For the first wire:
R1 = ρ1 * (L1/A1)
For the second wire:
R2 = ρ2 * (L2/A2)
Since the total resistance of the composite wire does not change with temperature, we can equate the total resistance of the composite wire to the sum of the resistances of the individual sections:
R_total = R1 + R2
Now, let's express the resistivity and temperature coefficient of resistivity in terms of the initial resistivity:
ρ1 = α1 * ρ_initial
ρ2 = α2 * ρ_initial
Substituting these values into the resistance equations, we get:
R1 = α1 * ρ_initial * (L1/A1)
R2 = α2 * ρ_initial * (L2/A2)
Substituting these expressions for R1 and R2 into the equation for the total resistance, we get:
R_total = α1 * ρ_initial * (L1/A1) + α2 * ρ_initial * (L2/A2)
Since the total resistance of the composite wire is constant, we can cancel out the ρ_initial term from both sides of the equation:
α1 * (L1/A1) + α2 * (L2/A2) = 0
Now, let's simplify the equation further:
α1 * (L1/A1) = - α2 * (L2/A2)
Dividing both sides of the equation by α1 * α2 * L2 * A1, we get:
(L1/A1) / (L2/A2) = - α2 / α1
Substituting the given values of α1 and α2 into this equation, we can solve for the ratio of the lengths:
(L1/A1) / (L2/A2) = - (-0.000670) / 0.00603
Simplifying further, we get:
(L1/A1) / (L2/A2) = 0.111
Therefore, the ratio of the length of the first section to the length of the second section is 0.111.
To find the ratio of the length of the first section to the length of the second section, we can use the formula for the resistance of a wire:
R = ρ * L / A,
where R is the resistance, ρ is the resistivity, L is the length, and A is the cross-sectional area.
Let's assume that the lengths of the first and second sections are L1 and L2, respectively.
The total resistance of the composite wire is given by:
R_total = R1 + R2,
where R1 is the resistance of the first section and R2 is the resistance of the second section.
The resistance of the first section, R1, can be expressed as:
R1 = ρ1 * L1 / A,
where ρ1 is the resistivity of the first section.
The resistance of the second section, R2, can be expressed as:
R2 = ρ2 * L2 / A,
where ρ2 is the resistivity of the second section.
According to the problem, the total resistance of the composite wire does not change with temperature. This means that the ratio of the resistances of the first and second sections is equal to the ratio of the lengths:
R1 / R2 = L1 / L2.
Substituting the expressions for R1 and R2, we get:
(ρ1 * L1 / A) / (ρ2 * L2 / A) = L1 / L2.
Canceling out the common terms, we can simplify the equation to:
ρ1 / ρ2 = L1 / L2.
Given the values of ρ1, ρ2, and their respective temperature coefficients α1 and α2, we can find the ratio of the lengths:
α1 / α2 = (L1 - L1₀) / (L2 - L2₀),
where L1₀ and L2₀ are the initial lengths of the first and second sections, respectively.
Since L1₀ = L2₀, we can simplify the equation to:
α1 / α2 = L1 / L2.
Substituting the values of α1, α2, L1, and L2, we can solve for the ratio:
0.00603 / -0.000670 = L1 / L2.
Simplifying the equation, we find:
L1 / L2 = -0.00603 / 0.000670.
Therefore, the ratio of the length of the first section to the length of the second section is approximately -9.007.
(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.