Efficiency Formula: Derivation and Solved Examples

• It relates to how close we can get to a certain output with the supplied input with as minimal wastage as possible.
• Efficiency is the ability to minimize the waste of materials, efforts, energy, and time in performing a task or delivering the desired result.
• It can be calculated as the ratio of usable output to total input.
• The efficiency of a device is defined as the ratio of energy transferred to a usable form to the total energy provided initially.

Key Terms: Efficiency, Efficiency formula, Work, Energy, Power, Heat, Total energy, Input energy, Output energy, Ratio, Percentage, Fraction, SI units

What is Efficiency?

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Efficiency is defined as the ratio of work done by a machine with respect to the total energy it has expended.

• Simply, the efficiency formula is a measurement of how much work or energy we are supposedly conserving during the procedure.
• Efficiency is clearly a distinction between the output of the energy to the input of the energy in any system or machine.
• For example, let’s say a heat engine can convert 80 percent of the fuel it uses into motion.
• Then at least 20 percent of the fuel may be dissipated as heat throughout the process.
• This process is a typical example of how you can compare the output of energy to the input in any machine.
• Theoretically, any process which is deemed perfect is known to have an efficiency of 100%.
• In other words, you can justify efficiency as a process where we tend to get close to an individual outcome of an already provided input while creating as much less wastage as possible.
• It decides the ability to deplete wasting resources, efforts, or energy during the performance of something to generate a likewise result.
• Efficiency is denoted by η.

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What is the Formula of Efficiency?

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The efficiency formula is generally confused with effectiveness. They are totally different terms if considered according to their literary outlooks. Efficiency is when things are known to be done accurately, while effectiveness is when things are pursued according to how they were expected.

The efficiency formula can be given as

For energy:

\(\eta = \frac{\text{Energy output}}{\text{Energy input}}\times 100\%\)

For work: 

\(\eta = \frac{\text{Work output}}{\text{Work input}}\times 100\%\)

The efficiency formula is generally seen to evaluate the efficiency of any given input of a particular machine or system. In a nutshell, efficiency has no unit and can be expressed in the form of a percentage.

Derivation of Efficiency Formula

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As per the efficiency formula, Energy can be denoted as: 

η = Energy output/Energy input × 100%; 

while Work can be denoted as: 

η = W_out/W_in × 100%, respectively. 

For a clearer understanding, it is better to know what the terms mean when writing the formula.

• Quite clearly, η is usually referred to as efficiency. It is a Greek letter, known as “eta”.
• W_out is termed as the amount of work or energy that is witnessed to be produced in a process. Its unit is Joules (J).
• Similarly, W_in is work or energy that is invested in a process that claims a resultant output. Its unit is Joules (J).

Solved Examples

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Ques. A heat energy of 800 J is produced as work done by a heat engine. If the efficiency of the engine is 40% then find the input energy supplied.

Ans. Given:

• Energy output = 800J
• Efficiency = 40%

The formula of Efficiency is given by

η = (Energy Output/Energy Input) × 100

⇒ Energy Input = (Energy Output/ η) x 100

⇒ Energy Input = 800/0.40

⇒ Energy Input = 2000 J

Ques. A cyclist on his bicycle puts 300 J of work and the useful work given out is 200 J, then what is the efficiency of the cyclist?

Ans. Given

• Work Output = 300 J
• Work Input = 200 J 

The formula of Efficiency is given by

η = (Energy Output/Energy Input) × 100

⇒ η = (200/300) × 100

⇒ η = 67%

Things to Remember

• The efficiency formula helps determine a quantitative value by obtaining the ratio of useful output with respect to the total input.
• For energy, the efficiency formula is given by, η = Energy output/Energy input × 100%
• For work, the efficiency formula is given by, η = W_out/W_in × 100%.
• Efficiency is also often confused with the meaning of effectiveness.
• They are totally different terms with distinctive literary perspectives.
• Efficiency is when things are known to be done accurately, while effectiveness is when things are pursued according to how they were expected.

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