**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 ‘

**’ for**

*p***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.

**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(Ï„) dÏ„`

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 Î© 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" "Î©`

`\"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.

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