Question

Electrical power $P$ is given by expression: $V \times x$. What is $x$?

Answer 1

To answer this question, we have to deduce the expression of the electric power in terms of voltage. For this we have to use the formula of the potential difference in terms of the work done which must be substituted into the formula of the electric power.

Formula used: The formulae used for solving this question are given by
$V = \dfrac{W}{q}$, here $V$ is the potential difference, $W$ is the work done, and $q$ is the charge.
$P = \dfrac{{dW}}{{dt}}$, here $P$ is the electric power, $W$ is the work done, and $t$ is the time.

Complete step by step answer
We know that the potential difference is equal to the work done in bringing a unit positive charge from the lower to the higher potential. So the potential difference can be given by
$V = \dfrac{W}{q}$
$\Rightarrow W = qV$ (1)
Now, we know that the power is the rate of work done. This means that
$P = \dfrac{{dW}}{{dt}}$ (2)
So for getting the electric power, we substitute (1) in (2)
$P = \dfrac{{d\left( {qV} \right)}}{{dt}}$
For a given value of the potential difference, we can take it to be constant. So we have
$P = V\dfrac{{dq}}{{dt}}$ (3)
Now, we know that the current is defined as the rate of flow of charge through a conductor, which means that
$i = \dfrac{{dq}}{{dt}}$
So (3) becomes
$P = V \times i$ (4)
According to the question, the electrical power is given as
$P = V \times x$ (5)
Comparing (4) and (5) we get
$x = i$
Hence, $x$ is the electric current.

Note
The electric power has three expressions. One is in terms of the voltage and the current, the second is in terms of the voltage and the resistance, and the third in the terms of the current and the resistance. The two of these three can be derived from any one of them using the Ohm’s law.