Question

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 \(\rho_{\text {ard }}=40.0 \cdot 10^{-8} \Omega \mathrm{m}\) ); the other is made of copper (with resistivity \(\rho_{\text {copper }}=1.68 \cdot 10^{-5} \Omega \mathrm{m}\) ). a) Calculate the ratio of the power dissipated by the two wires, \(P_{\text {copper }} / P_{\text {sted }}\) when they are connected in parallel and a potential difference of \(V=100 . V\) 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?

Short answer

Question: Calculate the ratio of power dissipated in a copper wire to that in a steel wire when they are connected in parallel with a potential difference applied. Explain the reason for using copper over steel in power transmission lines. Answer: The ratio of power dissipated in the copper wire to that in the steel wire can be calculated using the formula: $$\frac{P_{copper}}{P_{steel}} = \frac{I_{copper}^2 R_{copper}}{I_{steel}^2 R_{steel}}$$. The reason for using copper over steel in power transmission lines is that copper presents less resistance and dissipates less power as heat, making it more efficient for power transmission.

Step-by-step solution

Find the resistance of each wire

We have the resistivity, length, and cross-sectional area for each wire, so we can calculate their respective resistances using the formula: $$R = \frac{\rho L}{A}$$ Where \(\rho\) is the resistivity, \(L\) is the length, and \(A\) is the cross-sectional area. The area of a circle can be found using the formula: $$A = \pi r^2$$ So we have: $$R_{steel} = \frac{\rho_{steel} L}{\pi r^2}$$ $$R_{copper} = \frac{\rho_{copper} L}{\pi r^2}$$

Calculate the individual currents through each wire

When connected in parallel, the potential difference across both the wires remains the same (\(V = 100 V\)). We can use Ohm's law to find the current through each wire: $$I = \frac{V}{R}$$ So we have: $$I_{steel} = \frac{V}{R_{steel}}$$ $$I_{copper} = \frac{V}{R_{copper}}$$

Calculate the power dissipation for each wire

Now, we can calculate the power dissipation in each wire using the formula: $$P = I^2R$$ So we have: $$P_{steel} = I_{steel}^2 R_{steel}$$ $$P_{copper} = I_{copper}^2 R_{copper}$$

Calculate the power dissipation ratio

Now, we can calculate the ratio of power dissipation in the copper wire to that in the steel wire as follows: $$\frac{P_{copper}}{P_{steel}} = \frac{I_{copper}^2 R_{copper}}{I_{steel}^2 R_{steel}}$$

Explain the reason for using copper over steel in power transmission lines

After calculating the ratio of power dissipation in the copper wire to that in the steel wire, we can see that less power is dissipated in the copper wire compared to the steel wire. This means that more power can be transmitted along the copper wire, making it a more efficient material for power transmission lines. Additionally, copper has less resistance, which results in less energy loss in the form of heat, further enhancing the efficiency of power transmission.