KAHIBARO
Table of Contents
Meaning of Efficiency
In physics, efficiency tells us how well a system converts input energy or input power into useful output. No real machine converts all input into useful work, because some energy is usually transferred in unwanted ways, often as heat, sound, or frictional losses.
Efficiency is a ratio, so it compares what we want to get out with what we have to put in.
If a machine receives energy $E_{\text{in}}$ and delivers useful energy $E_{\text{useful}}$, then its efficiency is
$$
\eta = \frac{E_{\text{useful}}}{E_{\text{in}}}
$$
If we are looking at power instead of total energy, then
$$
\eta = \frac{P_{\text{useful}}}{P_{\text{in}}}
$$
Efficiency is often written using the Greek letter eta, $\eta$.
Important formulas for efficiency:
$$
\eta = \frac{E_{\text{useful}}}{E_{\text{in}}}
$$
$$
\eta = \frac{P_{\text{useful}}}{P_{\text{in}}}
$$
Percentage efficiency:
$$
\eta_{\%} = \eta \times 100\%
$$
For any real machine:
$$
0 \le \eta \le 1
$$
or equivalently,
$$
0\% \le \eta_{\%} \le 100\%
$$
Why Efficiency Matters
Efficiency helps us judge the performance of engines, motors, light bulbs, batteries, power plants, and even the human body. Two devices may produce the same useful result, but the more efficient one needs less input energy.
A high efficiency means less wasted energy. A low efficiency means a large fraction of the input is lost to forms that are not useful for the intended task.
For example, an electric motor may take in electrical power and convert most of it into mechanical power, but some of the energy becomes heat in the wires and moving parts. The output is useful mechanical power, while the heating is usually unwanted.
Efficiency as a Percentage
Because efficiency is a ratio, it has no unit. It is common to express it as a percentage:
$$
\text{Efficiency percentage} = \frac{\text{useful output}}{\text{input}} \times 100\%
$$
If a machine has efficiency $0.80$, we say it is $80\%$ efficient.
The table below shows some simple interpretations.
| Efficiency | Percentage | Meaning |
|---|---|---|
| $1.00$ | $100\%$ | All input becomes useful output |
| $0.75$ | $75\%$ | 75 percent useful, 25 percent wasted |
| $0.40$ | $40\%$ | Less than half the input is useful |
| $0.10$ | $10\%$ | Most input is wasted |
In practice, a value of exactly $100\%$ is an ideal limit for ordinary machines.
Energy Efficiency and Power Efficiency
Sometimes we care about the total energy used over a whole process. Sometimes we care about the rate at which energy is transferred. This leads to two closely related forms of efficiency.
Energy efficiency compares useful output energy with total input energy. This is useful when discussing a complete task, such as lifting a load or charging a battery.
Power efficiency compares useful output power with input power. This is useful when the system runs continuously, such as a motor or generator.
If the machine operates steadily over a time interval and both input and output are measured over the same interval, the two forms give the same numerical value, because
$$
P = \frac{E}{t}
$$
so
$$
\eta = \frac{P_{\text{useful}}}{P_{\text{in}}}
= \frac{E_{\text{useful}}/t}{E_{\text{in}}/t}
= \frac{E_{\text{useful}}}{E_{\text{in}}}
$$
Useful Output and Wasted Energy
A key idea is that not all output is useful. Energy is conserved, but only part of it may go into the desired effect.
If the total input energy is split into useful energy and wasted energy, then
$$
E_{\text{in}} = E_{\text{useful}} + E_{\text{wasted}}
$$
This means efficiency can also be written as
$$
\eta = 1 - \frac{E_{\text{wasted}}}{E_{\text{in}}}
$$
This form is helpful when the wasted part is easier to estimate than the useful part.
Efficiency does not tell us whether energy disappears. Energy is always conserved.
Efficiency tells us what fraction of the input becomes useful for the intended purpose.
Simple Examples
Suppose a machine takes in $500 \, \text{J}$ of energy and delivers $350 \, \text{J}$ of useful work. Then
$$
\eta = \frac{350}{500} = 0.70
$$
So the efficiency is
$$
70\%
$$
The wasted energy is
$$
500 - 350 = 150 \, \text{J}
$$
As another example, a motor receives input power $200 \, \text{W}$ and delivers useful mechanical power $150 \, \text{W}$. Then
$$
\eta = \frac{150}{200} = 0.75
$$
So the motor is $75\%$ efficient.
Rearranging the Formula
The efficiency equation can be rearranged to find unknown quantities.
If efficiency and input are known, useful output is
$$
E_{\text{useful}} = \eta E_{\text{in}}
$$
or
$$
P_{\text{useful}} = \eta P_{\text{in}}
$$
If efficiency and useful output are known, input is
$$
E_{\text{in}} = \frac{E_{\text{useful}}}{\eta}
$$
or
$$
P_{\text{in}} = \frac{P_{\text{useful}}}{\eta}
$$
These forms are often used in practical calculations.
Common Situations
Many everyday devices can be understood using efficiency.
A lamp converts electrical energy into light and heat. If the intended output is light, then only the light part counts as useful.
A car engine converts chemical energy in fuel into mechanical energy of motion, but a large amount becomes heat and sound.
A crane uses input power from a motor to raise a load. The useful output is the rate at which the load gains gravitational potential energy.
The table below shows the basic structure.
| Device | Input | Useful output | Common wasted forms |
|---|---|---|---|
| Electric motor | Electrical energy | Mechanical energy | Heat, sound |
| Light bulb | Electrical energy | Light | Heat |
| Car engine | Chemical energy | Mechanical energy | Heat, sound |
| Generator | Mechanical energy | Electrical energy | Heat, sound |
Visual Idea
A simple energy flow picture helps show the meaning of efficiency.
Important Limits
Efficiency cannot be negative, and for ordinary machines it cannot be greater than $1$.
If someone calculates an efficiency greater than $100\%$, that usually means there is a mistake in the calculation, in the units, or in identifying the useful output and total input.
Always check that
$$
\eta \le 1
$$
If your answer is greater than $1$, recheck the numbers, units, and definitions of input and useful output.
Final Idea
Efficiency is a simple but powerful measure of performance. It tells us how much of the energy or power supplied to a system becomes useful output. In physics, this helps us compare machines, understand energy losses, and calculate the input needed to achieve a desired result.