A coal-fired power station produces 600 MW of electrical power. The electricity is transmitted to distant towns and cities through overhead cables. The stat...

Marking Scheme

• (a) \(I = \frac{P}{V} = \frac{600\,000\,000}{132\,000}\) [1]; \(I = 4545\text{ A}\) [1]
• (b) \(P_{\text{loss}} = I^2 R = 4545^2 \times 4.0\) [1]; \(P_{\text{loss}} = 82\,600\,000\text{ W} \approx 83\text{ MW}\) [1]
• (c) \(I = \frac{600\,000\,000}{400\,000} = 1500\text{ A}\); \(P_{\text{loss}} = 1500^2 \times 4.0\) [1]; \(P_{\text{loss}} = 9\,000\,000\text{ W} = 9.0\text{ MW}\) [1]
• (d) For fixed power, increasing voltage reduces current (since \(P = IV\)) [1]; \(P_{\text{loss}} = I^2 R\) means loss depends on the square of current, so a smaller current gives a much larger reduction in power lost [1]

Explanation

(a) The power station produces 600 MW = 600,000,000 W. Using \(P = IV\):

\(I = \frac{P}{V} = \frac{600\,000\,000}{132\,000} = 4545\text{ A}\)

(b) The power dissipated as heat in the cables depends on the current and resistance:

\(P_{\text{loss}} = I^2 R = (4545)^2 \times 4.0 = 20\,657\,025 \times 4.0 = 82\,600\,000\text{ W} \approx 83\text{ MW}\)

This is about 14% of the total power output - a substantial waste.

(c) At the higher transmission voltage:

\(I = \frac{600\,000\,000}{400\,000} = 1500\text{ A}\)

\(P_{\text{loss}} = (1500)^2 \times 4.0 = 2\,250\,000 \times 4.0 = 9\,000\,000\text{ W} = 9.0\text{ MW}\)

This is only about 1.5% of the total power - a dramatic improvement.

(d) The key insight is the squared relationship in \(P_{\text{loss}} = I^2 R\). When the voltage is increased from 132 kV to 400 kV (a factor of about 3), the current decreases by the same factor of 3 (from 4545 A to 1500 A). But because power loss depends on \(I^2\), the loss decreases by a factor of \(3^2 = 9\). This is confirmed by the numbers: 83 MW / 9.0 MW is approximately 9. This squared dependence is the fundamental reason why electricity is transmitted at very high voltages - even a modest increase in transmission voltage produces a large reduction in energy wasted as heat in the cables.