Power & the Grid

Power & the Grid

Every appliance transfers energy at a certain rate: its power, measured in watts. Four short equations cover the whole topic. They also answer a question worth asking. Electricity crosses the country on cables at hundreds of thousands of volts, which sounds like the most dangerous possible way to do it. This module works out why it is done that way, with numbers.

The words for it

Two power equations and two energy equations. Knowing which to reach for matters as much as knowing them:

Find the power

A phone charger works at 12 V and draws a current of 3 A. What is its power, in watts (W)?

Where the second one comes from

P = I² R is not a separate fact. It falls straight out of the two equations you already have. Start from P = V I. For a resistance, Ohm's law says V = I R. Substitute that in: P = (I R) × I = I² R Which is worth knowing for two reasons. It is one fewer thing to remember, and it makes the behaviour obvious: because the current is squared, doubling the current does not double the heat wasted in a cable, it quadruples it. That single fact is the reason the National Grid is built the way it is.

Find the heating power

A heating element of resistance 5 Ω carries a current of 2 A. What power does it dissipate as heat, in watts (W)?

Energy from power and time

A 100 W lamp is left switched on for 30 seconds. How much energy does it transfer, in joules (J)?

Which equation, and what answer?

A charge of 5 C is moved through a potential difference of 12 V. Which equation applies, and what is the energy transferred?

• E = Q V, giving 60 J, because you have been given a charge and a potential difference and neither a time nor a current
• E = P t, giving 60 J, because energy always comes from power multiplied by time
• P = V I, giving 60 W, because voltage multiplied by the other quantity gives power
• None of them: you would also need to know how long the charge took to move

Why hundreds of thousands of volts

Suppose a power station sends 1 MW along a cable whose total resistance is 5 Ω. Since P = V I, the current needed depends entirely on the voltage chosen. At 1000 V the cable must carry 1,000,000 ÷ 1000 = 1000 A. The heat wasted is I² R = 1000² × 5 = 5,000,000 W, which is five times the power being sent. The scheme is not merely inefficient, it is impossible. At 100,000 V the same power needs only 1,000,000 ÷ 100,000 = 10 A. Now the loss is 10² × 5 = 500 W, which is a rounding error on a megawatt. Raising the voltage by a factor of 100 cut the current by 100, and because the current is squared it cut the waste by 10,000. That is the entire argument for high-voltage transmission, and it is why the voltage is stepped back down before it reaches anybody.

Label the National Grid

Here is the journey of electricity across the National Grid. Drag each label onto the correct part.

Ten times the voltage

A grid operator raises the transmission voltage by a factor of ten, delivering the same power along the same cables. What happens to the energy wasted heating those cables?

• It falls to about a hundredth of what it was, because the current falls to a tenth and the loss depends on the current squared
• It falls to about a tenth, in proportion to the change in voltage
• It stays the same, because the same power is being delivered either way
• It rises to about a hundred times as much, because higher voltage means more energy in the cables

Up, then down

Transformers appear twice on the journey, doing opposite jobs for different reasons:

Which equation would you use?

Which are true?

Select ALL THREE statements that are TRUE.

• P = I² R follows from P = V I together with V = I R, so it is not a separate fact to memorise
• Raising the transmission voltage by a factor of ten cuts the heating loss in the cables by a factor of about a hundred
• Transformers are used at both ends of the grid, and the one nearest homes steps the voltage down
• High transmission voltage is used to make the electricity travel faster along the cables
• A high current in the cables reduces the heat lost from them
• E = P t gives an answer in joules whether the time is in seconds or minutes

Power summary

The power of an appliance is P = V I, and substituting V = I R gives the power wasted heating a resistance, P = I² _____. Energy transferred is E = P _____, with the time in seconds, or E = Q V. The National Grid transmits at high voltage so that the _____ is low, and because the loss depends on that quantity _____, a small reduction in it saves a great deal of energy. A step-down transformer then lowers the voltage for homes.