What is Power? – Definition, Formula, Unit and Examples

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Unit of Power and Commercial Unit of Energy

Suppose an old man takes 10 minutes to do a particular “work” whereas a young man takes only 5 minutes to do the same work. It is obvious that the “rate of doing work” of the young man is more than that of the old man. The rate of doing work is known as power, so we can say that the power of young man is more than that of the old man.

Thus, power is defined as the rate of doing work. We can obtain power by dividing the ‘Work done’ by ‘Time taken’ for doing the work. That is,

Power = \(\frac{\text { Work done }}{\text { Time taken }}\)
or P = \(\frac{W}{t}\)
where P = power
W = work done
and t = time taken
In other words, power is the work done per unit time or power is the work done per second. Please note that the value which we get by dividing ‘Work done’ by ‘Time taken’ actually gives us ‘Average power’.

We know that when work is done, an equal amount of energy is consumed. So, we can also define power by using the term ‘energy’ in place of ‘work’. Thus, power is also defined as the rate at which energy is consumed (or utilised).

We can also obtain power by dividing ‘Energy consumed’ by ‘Time taken’ for consuming the energy. That is,
Power = \(\frac{\text { Energy consumed }}{\text { Time taken }}\)
or P = \(\frac{E}{t}\)
where P = power
E = energy consumed
and t = time taken
We can now say that: Power is the rate at which work is done or energy is consumed. It is clear from the above discussion that we can write the formula for calculating power in terms of ‘work done’ or in terms of ‘energy consumed’. Power is a scalar quantity which has only magnitude but no direction.

Units of Power

Power is obtained by dividing ‘work done’ by ‘time taken’ to do the work. Now, work is measured in the unit of ‘joule’ and the time is measured in the unit of ‘second’, so the unit of power is ‘joules per second’. This unit of power is called ‘watt’.

Thus, the SI unit of power is watt which is denoted by the symbol W. We can now define the unit of power ‘watt’ as follows : 1 watt is the power of an appliance which does work at the rate of 1 joule per second. We can also define watt by using the term ‘energy’ as follows : 1 watt is the power of an appliance which consumes energy at the rate of 1 joule per second. We can write an expression for watt as follows :
1 watt = \(\frac{1 \text { joule }}{1 \text { second }}\)
or 1 W = \(\frac{1 \mathrm{~J}}{1 \mathrm{~s}}\)
So 1 watt = 1 joule per second

Watt is an important unit of power since it is used in electrical work. The power of an electrical appliance tells us the rate at which electrical energy is consumed by it. For example, a bulb of 60 watts power consumes electrical energy at the rate of 60 joules per second (60 J/s or 60 J s^-1).

Different electrical appliances have different power ratings. The greater the power of an appliance, and the longer it is switched on for, the more electrical energy it consumes. The unit of power called ‘watt’ is named after a Scottish inventor, engineer and designer James Watt who became famous for improving the design of steam engine.

Watt is a small unit of power. Sometimes bigger units of power called kilowatt (kW) and megawatt (MW) are also used.
1 kilowatt = 1000 watts
or 1 kW = 1000 W
And 1 megawatt = 1000,000 watts
or 1 MW = 1000,000 W
or 1 MW = 10⁶ W
A yet another unit of power is called ‘horse power’ (h.p.) which is equal to 746 watts. Thus,
1 horse power = 746 watts
or 1 h.p. = 746 W
This means that 1 horse power is equal to about 0.75 kilowatt (0.75 kW).

The unit called ‘horse power’ originated long back when steam engines first replaced ‘horses’ as a source of power. These days the powers of engines (of cars, and other vehicles, etc.) are expressed in the unit called ‘brake horse power’ (b.h.p.).

Brake horse power is the unit of power equal to one horse power which is used in expressing power available at the shaft of an engine. The b.h.p. of Maruti-800 car is 37 whereas that of Maruti Zen is 60. The more powerful a car is, the quicker it can accelerate or climb a hill, that is, more rapidly it does work. We will now solve some problems based on power.

Example Problem 1.
A body does 20 joules of work in 5 seconds. What is its power ?
Solution:
Power is calculated by using the formula :
Power = \(\frac{\text { Work done }}{\text { Time taken }}\)
Here, Work done = 20 J
And, Time taken = 5 s
So, putting these values in the above formula, we get:
Power = \(\frac{20 \mathrm{~J}}{5 \mathrm{~s}}\)
= 4 J/s
Thus, Power = 4 W (because 1 J/s = 1 W)
Thus, the power of this body is 4 watts.

Example Problem 2.
What is the power of a pump which takes 10 seconds to lift 100 kg of water to a water tank situated at a height of 20 m ? (g = 10 m s^-2)
Solution:
In this problem first of all we have to calculate the work done by the pump in lifting the water against the force of gravity. We know that the work done against gravity is given by the formula :

W = m × g × h
Here, Mass of water, m = 100 kg
Acceleration due to gravity, g = 10 m s^-2
And, Height, h = 20 m
So, putting these values in the above formula, we get:
Work done, W = 100 × 10 × 20
= 20000 J
And, Time taken, t = 10 s
Now, we know that:
Power, P = \(\frac{W}{t}\)
= \(\frac{20000}{10}\)
= 2000 watts (or 2000 W)
Thus, the power of this pump is 2000 watts. This power can be converted from watts into kilowatts by dividing it by 1000.
So, Power = \(\frac{2000}{1000}\) kilowatts
= 2 kilowatts (or 2 kW)

Example Problem 3.
An electric bulb consumes 7.2 kJ of electrical energy in 2 minutes. What is the power of the electric bulb ?
Solution:
We know that :
Power = \(\frac{\text { Energy consumed }}{\text { Time taken }}\)
Energy consumed = 7.2 kJ
= 7.2 × 1000 J
= 7200 J
And, Time taken = 2 minutes
= 2 × 60 seconds
= 120 s
Now, putting these values of ‘energy consumed’ and ‘time taken’ in the above formula, we get:
Power = \(\frac{7200 \mathrm{~J}}{120 \mathrm{~s}}\)
= 60 J/s
= 60 W
Thus, the power of this electric bulb is 60 watts.

Commercial Unit of Energy

The commercial unit (or trade unit) of energy is kilowatt-hour which is written in short form as kWh. Kilowatt-hour is usually used as a commercial unit of electrical energy. This is discussed below.

The SI unit of electrical energy is joule and we know that “A joule is the amount of electrical energy consumed when an appliance of 1 watt power is used for one second”. Actually, joule represents a very small quantity of energy and, therefore, it is inconvenient to use where a large quantity of energy is involved.

So, for commercial purposes we use a bigger unit of electrical energy which is called “kilowatt-hour”. One kilowatt-hour is the amount of electrical energy consumed when an electrical appliance having a power rating of 1 kilowatt is used for 1 hour.

Since a kilowatt means 1000 watts, so we can also say that one kilowatt-hour is the amount of electrical energy consumed when an electrical appliance of 1000 watts is used for 1 hour.

Relation Between Kilowatt-Hour and Joule

1 kilowatt-hour is the amount of energy consumed at the rate of 1 kilowatt for 1 hour. That is,
1 kilowatt-hour = 1 kilowatt for 1 hour
or 1 kilowatt-hour = 1000 watts for 1 hour ……………(1)
But : 1 watt = \(\frac{1 \text { joule }}{1 \text { second }}\)
So, equation (1) can be rewritten as :
1 kilowatt-hour = 1000 \(\frac{\text { joules }}{\text { seconds }}\) for 1 hour
And, 1 hour = 60 × 60 seconds
joules
So, 1 kilowatt-hour = 1000 \(\frac{\text { joules }}{\text { seconds }}\) × 60 × 60 seconds
or 1 kilowatt-hour = 36,00,000 joules (or 3.6 × 10⁶ J)
From this discussion we conclude that 1 kilowatt-hour is equal to 3.6 × 10⁶ joules of electrical energy. It should be noted that watt or kilowatt is the unit of electrical power but kilowatt- hour is the unit of electrical energy.

The electrical energy used in homes, shops, and industries is measured in kilowatt-hours (kWh). The electricity meter installed in our home records the electrical energy consumed by us in kilowatt-hours.

1 kilowatt-hour (or 1 kWh) of electrical energy is commonly known as ‘1 unit’ of electricity. Our electricity bill shows the electrical energy consumed by our household in a month in ‘kilowatt-hours’ or ‘units’ of electricity.

One unit of electricity (or 1 kWh) may cost anything from ₹ 3 to ₹ 5 (or even more). The rates of electrical energy vary from place to place and keep on changing from time to time. We will now solve some problems based on commercial unit of energy.

Example Problem 1.
A radio set of 60 watts runs for 50 hours. How many ‘units’ (kWh) of electrical energy are consumed ?
Solution:
We want to calculate the electrical energy in kilowatt-hours, so first we should convert the power of 60 watts is :
Power, P = 60 watts
= \(\frac{60}{1000}\) kilowatts
= 0.06 kW
And, Time, t = 50 h
Now, putting P = 0.06 kW and t = 50 h in the formula :
Power = \(\frac{\text { Energy consumed }}{\text { Time taken }}\)
We get: 0.06 kW = \(\frac{\text { Energy consumed }}{50 \mathrm{~h}}\)
So, Energy consumed = 0.06 kW × 50 h
= 3 kWh
Thus, the electrical energy consumed is 3 kWh or 3 units.

Example Problem 2.
A family uses 250 units of electrical energy during a month. Calculate this electrical energy in joules.
Solution:
250 units of electrical energy means 250 kWh of electrical energy. Now, we know that:
1 kWh of energy = 3.6 × 10⁶ J
So, 250 kWh of energy = 3.6 × 10⁶ × 250 J
= 9 × 10⁸ J
Thus, 250 units of electrical energy is equal to 9 × 10⁸ joules.