Electric Power and Electrical Energy

In this article, we will study electric power and electrical energy along with their definition, formula, and unit of measurement and solved numerical examples.

In practical applications, we are required to know how much electric power an electrical device can handle. From our practical experiences, we know that a 200 Watt bulb produces more light than a bulb of 100 Watt. Also, we know that electric utility companies charge the electricity bill for the electrical energy consumed by us over a certain period of time. Therefore, these two practices prove that the calculations of electric power and electrical energy are very important in circuit analysis.

What is Electric Power?

The rate at which work is done in an electrical circuit is known as electric power. In other words, electric power can be defined as the time rate of absorbing or expanding the electrical energy in an electric circuit.

The electrical power is denoted by the symbols ‘P’ for average power and ‘p’ for instantaneous power. The SI unit of electric power is Watt, denoted by W.

Mathematically, the electric power is given by the work done divided by time, i.e.

`\p={dw}/dt`

Derivation of Electric Power

In order to obtain the expression of electric power, i.e. the power in terms of electrical quantities such as voltage, current, resistance, etc., we consider the following circuit.

When a voltage is applied to this electric circuit it causes an electric current to flow through it. Therefore, work is done in flowing the current (or moving the charge) in the circuit. This work done in flowing the current or moving the charge per unit of time is known as electric power.

Hence, referring to the circuit, we get,

`\V="Voltage in volts"`

`\I="Current in Amperes"`

`\t="Time in seconds"`

Since, by the definition of voltage and current, we have,

`\V=("Work done "(W))/("Charge "(Q) )`

Therefore, the work done in the electric circuit is given by,

`\W=V×Q`

Also, by the definition of electric current, we have,

`\I=("Charge "(Q))/("Time "(t) )`

`\⟹Q=It`

Now, again by the definition of electric power, we have

`\P=W/t=(V×Q)/t`

`\⟹P=(V×It)/t`

`\∴P=VI"   Watts"`

Thus, we can see that the electric power in an electric circuit is simply the product of the voltage across the circuit element and the current through it.

Also, by Ohm’s law, we know that

`\V=IR`

Or

`\I=V/R`

Hence, by substituting these values of voltage and current in the expression of electric power, we get,

`\P=VI=I^2 R=V^2/R`

These three formulae are equally valid for the calculation of electrical power in any electric circuit.

Units of Electric Power

Since the electric power is given by,

`\P=W/t`

Where, the work done (W) is measured in Joules (J) and times (t) is measured in seconds, hence, the electric power can be measured in Joules per second (J/s). Where,

`\(1" Joule")⁄"second"=1" Watt"`

The other units of electric power are horsepower (H.P.), kilowatts (kW), and megawatts (MW).

Where,

`\1" h.p."=746" Watts"`

`\1" kW"=1000" Watts"`

`\1" MW"=10^6"  Watts"`

Passive Sign Convention for Power

The passive sign convention is the standard used to define whether the power is being absorbed or delivered by the element. The concept of passive sign convention is based on the current direction and voltage polarity for an element.

According to the passive sign convention, the power delivered or power absorbed by a circuit element can be defined as follows:

• If the electric current enters the circuit element at the positive terminal of the voltage and exits at the negative terminal, then the power is absorbed by the circuit element. The power absorbed is also called power dissipated or power received by the circuit element.

• If the electric current enters the circuit element at the negative terminal of the voltage and exits at the positive terminal, then the power is delivered by the element. The power delivered is also called power supplied by the element.

Note – In any electric circuit, the total electrical power supplied to the circuit must balance the total power absorbed, i.e.

Power delivered=-Power absorbed

What is Electrical Energy?

Energy is the capability to do work. In other words, the total amount of work done in an electric circuit is known as electrical energy. The electrical energy is usually denoted by the symbol ‘W’. The SI unit of electrical energy is Joule (J).

Mathematically, the electrical energy is given as the product of electric power and time for which the current flow through the circuit, i.e.

`\W=P×t`

`\∵P=VI=I^2 R=V^2/R`

Therefore, the electrical energy expended or absorbed in an electric circuit is given by,

`\W=VIt=I^2 Rt=(V^2 t)/R`

In the integral form, we can give the electrical energy as follows,

`\w(t)=∫_{t_1}^{t_2}p(τ) dτ`

I.e. the electrical energy over a time interval can be found by integrating the electrical power.

Unit of Electrical Energy

Since the electrical energy is given by the product of power and time, therefore, it can be measured in watt-seconds, where,

`\1" Ws"=1" Joule"`

The other units of electrical energy are Watt-hours (Wh), kilowatt-hours (kWh), and B.O.T. Unit (Board of Trade Unit).

Where,

`\1" Wh"=3600" Joules"`

`\1" kWh"=3.6×10^6"  joules"`

`\1" B.O.T. Unit"=1" kWh"`

Note – The electricity bill is made on the basis of the total amount of electrical energy consumed by the consumer over a certain period of time. The practical unit for charging electricity is 1 kWh. The 1 kWh is also known as B.O.T. (Board of Trade) unit or simply unit. Hence, when we say that we consumed 200 units of electricity, it means that electrical energy consumption is 200 kWh.

Applications of Power and Energy Formulae

The following description gives an idea about the applications of formulae of electric power and energy:

(1). The following formulas of electric power and electrical energy:

`\"Power",P=VI`

`\"Energy",W=VIt`

These formulas of electrical power and energy can be applied to any type of load such as a motor, bulb, heater, etc.

(2). The formulas of electric power and energy:

`\P=I^2 R=V^2/R`

`\W=I^2 Rt=(V^2 t)/R`

These formulas of electric power and electrical energy can be applied only to resistors and to devices such as electric bulbs, electric heaters, electric kettles, etc. where all the electrical energy consumed is transformed into heat.

Numerical Example – A resistor of 100 Ω has a DC voltage of 150 volts across it. Calculate, the power dissipated and energy consumed by the resistor in the 2 hours.

Solution – Given data,

`\"Resistance",R=100" "Ω`

`\"Voltage",V=150" "V`

`\"Time",t=2" hours"`

Now, the electrical power dissipated by the resistor is

`\P=V^2/R`

`\P=(150)^2/100=225" Watts"`

The electrical energy consumed is

`\W=P×t`

`\W=225×2`

`\∴W=450" Wh"`

Conclusion

Hence, in this article, we discussed about electric power, passive sign convention for power calculation, and electrical energy.