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?
Ah, the good old wires joining forces. Let's dive into this electrifying problem, shall we?
We know that the total resistance of the composite wire stays the same regardless of temperature changes. It means that the resistances of the two pieces must balance each other out.
Let's say the length of the first wire is L1, and the length of the second wire is L2. The resistivity of each wire can be calculated using the formula R = ρ * (L / A), where R is the resistance, ρ is the resistivity, L is the length, and A is the cross-sectional area.
For the first wire, the resistance is R1 = ρ1 * (L1 / A1).
For the second wire, the resistance is R2 = ρ2 * (L2 / A2).
Since the total resistance is constant, we have R1 + R2 = R2 + R1 (because addition is commutative, even in the wonderful world of wires).
Now, let's plug in the values and simplify this equation a bit:
(ρ1 * L1 / A1) + (ρ2 * L2 / A2) = (ρ1 * L2 / A2) + (ρ2 * L1 / A1)
We know that A1 = A2 (since the cross-sectional areas of the wires are the same), so let's get rid of that:
(ρ1 * L1) + (ρ2 * L2 / A2) = (ρ1 * L2) + (ρ2 * L1)
Now, let's isolate the length ratio (L1 / L2):
(ρ1 * L1) - (ρ2 * L1) = (ρ2 * L2) - (ρ1 * L2)
Factoring out L1 and L2:
L1 * (ρ1 - ρ2) = L2 * (ρ2 - ρ1)
Finally, dividing both sides by (ρ2 - ρ1):
L1 / L2 = (ρ2 - ρ1) / (ρ1 - ρ2)
Now plug in the values for ρ1, ρ2, α1, and α2, and calculate that ratio. I'll give you a moment to perform that mathematical magic!
By the way, don't worry, the Clown Bot is well-insulated from electric shocks.
To find the ratio of the length of the first section to the length of the second section, we need to use the concept of resistivity and temperature coefficient of resistivity.
Let's assume the length of the first wire section is L1 and the length of the second wire section is L2.
The resistance of a wire is given by the formula: R = ρ * (L / A), where R is the resistance, ρ is the resistivity, L is the length, and A is the cross-sectional area.
According to the problem, the total resistance of the composite wire does not change with temperature. This means that the total resistance of the two wire sections connected in series must be equal.
First, let's calculate the resistance of the first wire section:
R1 = ρ1 * (L1 / A), where ρ1 is the resistivity of the first wire.
Next, let's calculate the resistance of the second wire section:
R2 = ρ2 * (L2 / A), where ρ2 is the resistivity of the second wire.
Since the total resistance of the composite wire is the sum of the resistances of the two wire sections, we have:
R_total = R1 + R2
Since R_total does not change with temperature, we can write:
R1 + R2 = constant, or ρ1 * (L1 / A) + ρ2 * (L2 / A) = constant
Now, let's consider the temperature coefficient of resistivity:
ΔR / R = α * ΔT, where ΔR is the change in resistance, R is the original resistance, α is the temperature coefficient of resistivity, and ΔT is the change in temperature.
For the first wire section, we have:
ΔR1 / R1 = α1 * ΔT
Similarly, for the second wire section, we have:
ΔR2 / R2 = α2 * ΔT
Since both wire sections are at the same temperature, ΔT is the same for both sections. Therefore, we can write:
ΔR1 / R1 = ΔR2 / R2
Now, substitute the expressions for ΔR1 and ΔR2:
(R1 - constant) / R1 = (R2 - constant) / R2
Simplifying this equation, we get:
R1 * R2 - R1 * constant = R2 * R1 - R1 * constant
This simplifies to:
R1 * R2 = R2 * R1, which is always true.
From this, we can conclude that the ratio of the length of the first section to the length of the second section does not depend on the resistivities, but only on the cross-sectional area.
Therefore, the ratio of the lengths of the two wire sections is 1:1, or L1:L2 = 1:1.