To solve the problem of energy consumption in two uniform wires connected in series, we need to first determine their resistances and then calculate the power consumed in each wire. Given that one wire is twice the length of the other, we can use the properties of resistivity and the formula for resistance to find the answers.
Understanding the Problem
We have two wires made from the same material, each with a mass of \(10^{-3}\) kg. The longer wire has a length of 0.2 m, which means the shorter wire is 0.1 m long. The density of the material is \(11.34 \times 10^3\) kg/m³, and the resistivity is \(20.6 \times 10^{-8}\) ohm-m. The current flowing through both wires is 10 A.
Calculating the Cross-Sectional Area
First, we need to find the cross-sectional area of the wires. The mass \(m\) of a wire can be expressed as:
- m = density × volume
- Volume = cross-sectional area (A) × length (L)
Thus, we can rearrange this to find the cross-sectional area:
A = m / (density × L)
Finding the Area for Each Wire
For the longer wire (0.2 m):
A₁ = \( \frac{10^{-3}}{11.34 \times 10^3 \times 0.2} \) = \( \frac{10^{-3}}{2268} \) ≈ \( 4.41 \times 10^{-7} \) m²
For the shorter wire (0.1 m):
A₂ = \( \frac{10^{-3}}{11.34 \times 10^3 \times 0.1} \) = \( \frac{10^{-3}}{1134} \) ≈ \( 8.82 \times 10^{-7} \) m²
Calculating Resistance
The resistance \(R\) of a wire can be calculated using the formula:
R = \( \frac{\text{resistivity} \times \text{length}}{\text{cross-sectional area}} \)
For the longer wire:
R₁ = \( \frac{20.6 \times 10^{-8} \times 0.2}{4.41 \times 10^{-7}} \) ≈ \( 9.34 \times 10^{-2} \) ohms
For the shorter wire:
R₂ = \( \frac{20.6 \times 10^{-8} \times 0.1}{8.82 \times 10^{-7}} \) ≈ \( 2.33 \times 10^{-2} \) ohms
Calculating Power Consumption
The power \(P\) consumed by a resistor can be calculated using:
P = I²R
For the longer wire:
P₁ = \( 10^2 \times 9.34 \times 10^{-2} \) = 93.4 W
For the shorter wire:
P₂ = \( 10^2 \times 2.33 \times 10^{-2} \) = 23.3 W
Comparing Heat Generation
From our calculations, the longer wire consumes 93.4 W while the shorter wire consumes 23.3 W. Since power is directly related to heat generation, the longer wire will get significantly hotter due to its higher power consumption.
Summary of Findings
In summary, the rate of energy consumption in the longer wire is approximately 93.4 W, while the shorter wire consumes about 23.3 W. Therefore, the longer wire will get hotter due to its higher power dissipation.
