4.11 - Efficiency

Efficiency is the proportion transferred usefully

• A device is useful because it changes energy from one form into a useful form. But not all the input energy ends up there.
• Some energy is always wasted, usually changed into heat energy in the device and its surroundings.
• Efficiency measures how much of the input energy is transferred usefully rather than wasted. The less energy wasted, the more efficient the device.

Energy is conserved

None of the input energy disappears - it's split between useful and wasted output:

Total energy input = useful energy output + wasted energy

The total energy output always equals the total energy input. A more efficient device transfers a larger share to the useful form and wastes less.

No device is 100% efficient

• Because some energy is always wasted, no real device can transfer all of its input energy usefully.
• This means efficiency can never reach or go above 1 (or 100%).
• The wasted energy is usually changed into heat energy in the surroundings, where it's no longer useful.

Electric heaters are a special case

• Heaters are usually almost 100% efficient, because the energy they receive as electrical energy is meant to be changed into heat energy.
• The energy other devices waste as heat is the useful output of a heater.

Efficiency has no units

• Efficiency is a ratio of two energies (or two powers), so it has no units.
• It's written either as a decimal between 0 and 1, or as a percentage between 0% and 100%.

Formula for efficiency (energy)

Where:

• Useful energy output is the energy that does the intended job, in joules (J)
• Total energy input is the total energy supplied to the device, in joules (J)

This gives efficiency as a decimal. To express it as a percentage, multiply by 100. Because energy is conserved, total energy input equals total energy output, so either can be used as the denominator.

Worked example - Efficiency of a microwave oven

A microwave oven transfers 180,000 J of energy electrically from the mains. Of this, 110,000 J is transferred usefully to the thermal energy store of the food. Calculate the efficiency of the microwave oven.

Measuring the output power of a motor

• The output power of an electric motor can be measured by using it to lift a known mass through a known height at a constant speed.
• The useful work the motor does is the gravitational potential energy the mass gains as it rises.
• Dividing that work by the time taken gives the useful output power.

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Formula for output power

$$\text{Output power}=\frac{\text{work done}}{\text{time}}=\frac{mgh}{t}$$

Where:

• $m$ = Mass lifted, in kilograms (kg)
• $g$ = Gravitational field strength, in newtons per kilogram (N/kg)
• $h$ = Height the mass is raised, in metres (m)
• $t$ = Time taken, in seconds (s)

Gravitational field strength on Earth is about 9.8 N/kg (sometimes rounded to 10 N/kg or given as 9.81 N/kg). Always use the value given in the question.

Worked example - Efficiency of a motor lifting a load

A motor lifts a load of 70 N through a height of 150 cm. In doing so, 45 J of energy is wasted as heat and sound. Calculate the motor's efficiency.