Question

Derive the expression for power $P = VI$ in an electrical circuit.

Answer 1

Hint
According to Ohm's law at a constant temperature, the current flowing through a conductor will be directly proportional to the potential difference between the ends of the conductor. It is also inversely proportional to the resistance offered by the conductor. We can define the electrical power in a circuit as the amount of energy produced or absorbed in a circuit.
Ohm’s law gives that,
$\Rightarrow I = \dfrac{V}{R}$ (Where, $I$ stands for the current in a circuit, $V$ stands for the potential difference across the conductor, $R$ stands for the resistance offered by the conductor)
$\Rightarrow P = \dfrac{w}{t}$ (Where, $P$ stands for the power, $w$ stands for the work done and, $t$ stands for the time.)

Complete step by step answer
From Ohm’s law, we know that
$\Rightarrow I = \dfrac{V}{R}$
The definition of power says that it is the amount of electrical energy produced or consumed by a circuit. Also in other words we can say that the power in a circuit is the amount of work done by the circuit in unit time. It can be written as,
$\Rightarrow P = \dfrac{w}{t}$ … equation(A)
By the definition of potential difference we know that potential difference is the work done in bringing a charge from infinity to a particular point, we can write it as
$\Rightarrow V = \dfrac{w}{Q}$ (Where $V$ stands for the potential difference, $w$ stands for the work done and $Q$ stands for the total charge)
From the above equation, we get
$\Rightarrow w = VQ$
Substituting this expression of $w$ in equation (A), we get
$\Rightarrow P = \dfrac{{VQ}}{t}$ … equation(B)
Now, we know that by definition the current is defined as the charge flowing through a conductor per unit of time. That can be written as,
$\Rightarrow I = \dfrac{Q}{t}$
Substituting this value of current in equation (B), we get
$\Rightarrow P = VI$

Note
Power is measured using the unit watt. In most electrical devices we can see power ratings. Power rating will be given in the units of watts. It indicates the maximum amount of electrical energy converted into any other form of energy by the device. Some electrical devices such as electrical motors denote power in the units of horsepower.