In this article, I will explain the concept of electric power, its formula, different types, and solved numerical examples based on the power formula. In electric circuits, the rate of work done is referred to as electric power. Electric power is one of the fundamental electrical quantities in electric circuits. It plays a vital role in specifying the capability of an electric device. For example, as we know we use electric bulbs rated in watts like 60 W bulb or 100 W bulb. Therefore, understanding the concept of electric power is crucial for working with electrical devices.
What is Electric Power?
In an electric circuit, the rate of doing work in moving charge (electrons) is called electric power.
In other words, the amount of work done in an electric circuit per unit time is called electric power.
It is represented by the symbol “P” or “p”. Where, “P” is used to represent average power while “p” is used to represent instantaneous power i.e., power as the function of time.
Electric Power Formula:
Mathematically, if W joules is the amount of work done in an electric circuit in a time of t seconds. Then, the electric power in the circuit is given by,
$$P=\frac{W}{t}$$
In differential form,
$$p(t)=\frac{\text{d}w(t)}{\text{d}t}$$
SI Unit of Electric Power:
We can derive the unit of electric power directly from the above expression as follows.
$$\text{Unit of Power}=\frac{\text{Joules }(J)}{\text{Second }(s)}$$
Hence, the fundamental unit of electric power is Joules per second (J/s). This is also called Watt (W).
∴1 Watt (W) = 1 Joule per second
Where, Watt (W) is the SI unit of electric power.
In electrical power systems, we often use large units of electric power. Some commonly used large units of electric power are given below.
- kW (kilowatt) – 1 kW is equal to 1000 Watts or 103
- MW (Mega Watt) – 1 MW is equal to 106 W or 103
- GW (Giga Watt) – 1 GW is equal to 109 W or 103 MW or 106 kW.
Relation between Power, Voltage, and Current
In an electric circuit, electric power is related to the voltage and electric current. Here, is the mathematical explanation of how being power is related to the voltage and current.
When the voltage of V volts is applied to an electric circuit, it forces an electric current of I amperes through the circuit. As we know, electric current is nothing but the flow of electrons in the circuit. Therefore, voltage does the necessary work of W joules to move the electrons in the circuit.
Then, from the definition of voltage, we have,
$$V=\frac{W}{Q}$$
Where, Q is the charge in coulombs.
Also, we know, the relationship between electric charge (Q), current (I), and times (t) is given by,
$$Q=I × t$$
Therefore, the total amount of work done in moving the total charge (total number of electrons) in the circuit will be,
$$W=V×Q=V×It$$
As we know, electric power is the rate of work done in an electric circuit i.e.,
$$P=\frac{W}{t}=\frac{VIt}{t}$$
$$∴P=VI$$
Hence, electric power in an electric circuit is equal to the product of voltage and current in the circuit.
Relation between Power and Resistance
In an electric circuit, resistance is one of the fundamental circuit parameters. An electric resistor consumes the electric power and converts it into heat.
Hence, there is a relation between resistance and power which is explained below.
In an electric circuit power is given by,
$$P=VI$$
From Ohm’s law, we know that
$$V=IR$$
Where, R is the resistance of the circuit or conductor.
Therefore,
$$I=\frac{V}{R}$$
Substituting the values of voltage or current in the equation of power, we get,
$$P=IR×I=I^2 R$$
Also,
$$P=V×\frac{V}{R}=\frac{V^2}{R}$$
Hence, the relation between power and resistance is given by,
$$P=I^2 R=\frac{V^2}{R}$$
This formula of electric power can be used to calculate the power in a circuit when resistance and voltage or current are known.
Use of Power Formula
Case I – Electric power is given by,
$$P=VI$$
This formula of electric power can be used to calculate the electric power of any kind of load.
Case II – Electric power is also given by,
$$P=I^2 R=\frac{V^2}{R}$$
This formula of electric power can be used only for resistors and devices where all electric power is converted into heat, like in electric heaters, bulbs, kettles, etc.
Types of Electric Power
Based on the type of electric current, there are the following two types of electric power:
- DC Power
- AC Power
Let us discuss these two types of power in detail.
(1). DC Power:
When an electric circuit has a direct voltage and direct current, then, the electric power consumed in the circuit is called DC power.
DC power is measured as the product of direct voltage and direct current in the circuit i.e.,
$$P_{dc}=VI$$
DC power is produced by direct voltage sources like batteries, solar cells, dc generators. etc.
(2). AC Power:
When the applied voltage to the circuit is alternating voltage and the current flowing in the circuit is alternating current. Then, the power consumed in the circuit is called AC power.
AC power is further classified into the following types:
- Instantaneous Power
- Average Power
- Active Power
- Reactive Power
- Apparent Power
- Complex Power
Let us discuss each of these types of AC power in detail.
(a). Instantaneous Power:
In an AC circuit, the instantaneous power is defined as the electric power consumed at a particular instant of time.
Mathematically, the instantaneous power is given as the product of instantaneous voltage and current i.e.,
$$p=v.i$$
(b). Average Power:
In an AC circuit, the average power is defined as the electric power averaged over a complete cycle of alternating current.
In other words, the average of instantaneous electric power calculated over a complete cycle of alternating current is called average power.
Mathematically, the average power is given by,
$$P_{av}=\frac{1}{T} ∫_0^T p dt$$
Where, T is the time period of the alternating current and p is the instantaneous power.
(c). Active Power:
In an AC electric circuit, the part of total electric power that does some useful work in the circuit is called active power. It is represented by the letter “P” and is measured in Watts.
Mathematically, the active power is given as the product of RMS (Root Mean Square) values of voltage and current i.e.,
$$P=VI cosϕ$$
Where, V is the RMS value of AC voltage, I cosϕ is the active component of the circuit current, and ϕ is the power factor angle of the circuit.
(d). Reactive Power:
In an AC circuit, the part of total electric power that does not do any useful work, but flows back and forth and reacts upon itself is called reactive power.
It is denoted by the symbol Q and is measured in Var (Volt-Ampere Reactive).
Mathematically, the reactive power is given by,
$$Q=VI sinϕ$$
Where, V is the RMS value of voltage and I sinϕ is the reactive component of the circuit current, and ϕ is the power factor angle of the circuit.
(e). Apparent Power:
In an AC circuit, the total electric power supplied by the source to the load is called the apparent power. It is denoted by the symbol S and is measured in VA (Volt Ampere).
Mathematically, it is given as the product of the RMS value of AC voltage and current i.e.,
$$S=VI$$
It can also be given as the phasor sum of active power and reactive power i.e.,
$$S=P+jQ$$
(f). Complex Power:
In an AC circuit, the complex power is defined as the product of RMS voltage and the complex conjugate of the RMS current i.e.,
$$S=VI^*$$
The magnitude of the complex power is equal to that of the apparent power. It is also measured in VA.
Numerical Examples Based on Electric Power
Example (1) – In an electric circuit, if 500 joules of work is done in 2 minutes. Calculate the electric power consumed in the circuit.
Solution – Given data,
Work done (W) = 500 J
Time (t) = 2 minutes = 2 × 60 = 120 seconds
The electric power consumed in the circuit is given by,
$$P=\frac{W}{t}=\frac{500}{120}$$
∴P = 4.167 Watts
Example (2) – An electric current of 6 A is flowing through a resistor of 100 Ω. Calculate the power consumed by the resistor.
Solution – Given data,
Electric current (I) = 6 A
Resistance (R) = 100 Ω
The electric power consumed by the resistor will be,
$$P=I^2 R=6^2×100$$
∴P = 3600 W = 3.6 kW
Example (3) – An electric heater has a resistance of 2.5 kΩ and a voltage of 220 V is applied across it. What will be the power consumed by the heater?
Solution – Given data,
Resistance (R) = 2500 Ω = 2.5 kΩ
Voltage (V) = 220 V
The power consumption of the electric heater will be,
$$P=\frac{V^2}{R}=\frac{220^2}{2500}$$
∴P = 19.36 W
Example (4) – An electric motor draws a current of 16 A when it is connected to a supply of 200 volts. What will be the power rating of the electric motor?
Solution – Given data,
Current (I) = 16 A
Voltage (V) = 200 V
The power rating of the electric motor is,
P = V×I = 200 × 16
⸫ P = 3200 W = 3.2 kW
Example (5) – In an AC circuit, the voltage and current are given by,
$$v(t)=200 sin(314t+45)$$
$$i(t)=10 sin(314t+30)$$
Calculate, instantaneous power at t = 5 seconds.
Solution – Given data,
$$v(t)=200 sin(314t+45)$$
$$i(t)=10 sin(314t+30)$$
At t = 5 s, the instantaneous power will be,
The value of voltage and current at 5 seconds is,
$$v(5)=200 sin(314×5+45)=17.43 V$$
$$i(5)=10 sin(314×5+30)=3.42 A$$
Hence, the instantaneous power at t = 5 s is,
$$p(5)=v(5).i(5)=17.43×3.42$$
∴p(5) = 59.61 W
Example (6) – In an AC circuit, the RMS voltage and current are 220 V and 6 A respectively. If the power factor angle of the circuit is 37°. Then, calculate the active power, reactive power, and apparent power in the circuit.
Solution – Given data,
RMS voltage (V) = 220 V
RMS current (I) = 6 A
Power factor angle (ϕ) = 37°
The active power consumed in the circuit is,
$$P=VI cosϕ$$
$$⇒P=220×6×cos(37°)$$
∴P = 1054.19 W = 1.054 kW
The reactive power flowing in the circuit is,
$$Q=VI sinϕ$$
$$⇒Q=220×6×sin(37°)$$
∴Q = 794.39 VAr
The apparent power in the circuit is,
S = VI = 220×6
∴S = 1320 VA = 1.32 kVA
Conclusion
In conclusion, electric power is the time rate of work done in an electric circuit. Electric power is an important quantity in an electric circuit used to calculate the electricity consumed in the circuit. It is also used to analyze the behavior of an electric circuit.
In this article, I have explained electric power and concepts related to it. Also, I have included some solved numerical examples to explain the application of the electric power formula. Still, if you have any queries related to this topic, please let me know in the comment section. I will answer shortly.