Two conducting wires have identical lengths \(L_{1}=L_{2}=\) \(L=10.0 \mathrm{~km}\) and identical circular cross sections of radius \(r_{1}=r_{2}=r=1.00 \mathrm{~mm} .\) One wire is made of steel (with resistivity \(\rho_{\text {steel }}=40.0 \cdot 10^{-8} \Omega \mathrm{m}\) ); the other is made of copper (with resistivity \(\left.\rho_{\text {copper }}=1.68 \cdot 10^{-8} \Omega \mathrm{m}\right)\) a) Calculate the ratio of the power dissipated by the two wires, \(P_{\text {copper }} / P_{\text {steel }},\) when they are connected in parallel; a potential difference of \(V=100 . \mathrm{V}\) is applied to them. b) Based on this result, how do you explain the fact that conductors for power transmission are made of copper and not steel?

Step by step solution

01

Calculate the resistance of each wire

To calculate the resistance of a wire, we use the formula: \(R = \frac{\rho \cdot L}{A}\) where \(R\) is the resistance, \(\rho\) is the resistivity, \(L\) is the length of the wire, and \(A\) is the cross-sectional area. Since both wires have identical lengths and cross-sectional areas, we can calculate their respective resistances using the values given for their resistivities. For the steel wire: \(R_{steel} = \frac{\rho_{steel} \cdot L}{A_{steel}} = \frac{40.0 \cdot 10^{-8} \Omega m \cdot 10.0 \times 10^{3} m}{\pi \cdot (1.00 \times 10^{-3} m)^2} \) For the copper wire: \(R_{copper} = \frac{\rho_{copper} \cdot L}{A_{copper}} = \frac{1.68 \cdot 10^{-8} \Omega m \cdot 10.0 \times 10^{3} m}{\pi \cdot (1.00 \times 10^{-3} m)^2} \)

02

Calculate the equivalent resistance of the parallel connection

Now that we have the resistance of both wires, we can find the equivalent resistance of their parallel connection using the formula: \(\frac{1}{R_{eq}} = \frac{1}{R_{steel}} + \frac{1}{R_{copper}}\)

03

Find the total current through the parallel connection

Now, with the equivalent resistance, we can find the total current through the parallel connection using Ohm's law. The potential difference \(V = 100V\) is applied to the parallel connection: \(I_{total} = \frac{V}{R_{eq}}\)

04

Calculate the current through each wire

Since the wires are connected in parallel, the voltage across each wire is the same. We can use Ohm's law again to find the current through each wire: \(I_{steel} = \frac{V}{R_{steel}}\) \(I_{copper} = \frac{V}{R_{copper}}\)

05

Calculate the power dissipated by each wire

To calculate the power dissipated by each wire, we can use the formula: \(P = IV\) For the steel wire: \(P_{steel} = I_{steel} \cdot V\) For the copper wire: \(P_{copper} = I_{copper} \cdot V\)

06

Calculate the ratio of the power dissipated by the two wires

Now, we can find the ratio of the power dissipated by the copper wire to that of the steel wire: \(P_{copper}/P_{steel} = \frac{P_{copper}}{P_{steel}}\)

07

Explain the preference of copper over steel for power transmission

Based on the calculated power dissipation ratio, we can discuss why copper is preferred over steel for power transmission. Copper has a lower resistivity than steel, which means it has a lower resistance for the same length and cross-sectional area. Consequently, a copper conductor will allow more current to flow and dissipate less power as heat compared to a steel conductor. This makes copper more efficient for power transmission, as less energy is wasted as heat.

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