Question

Transmission lines transmit a voltage of $V$ volt to our houses from power stations, then the power $P$ supplied them is proportional to:
A. $\dfrac{1}{V}$
B. $V$
C. ${V^2}$
D. $\dfrac{1}{{{V^2}}}$

Answer 1

The rate of dissipation of electric energy is called electric power. This energy dissipated in time $t$ is given by the product of current set up in the conductor, the potential difference across the conductor and the time period. By using this formula, the relation between voltage and power can be found out.

Formula Used:
The energy dissipated $w$ in time $t$ is given by: $w = VIt$
where, $I$ is the current set up in the conductor, $V$ is the potential difference across the conductor and $t$ is the time.

Complete step by step answer:
The transmission lines transmit a voltage of $V$ volt to our houses from power stations. When a current $I$ is set up in a conductor across which a potential difference $V$ is applied, the energy dissipated in time $t$ is given by,
$w = VIt$
Therefore, the electrical power dissipated will be,
$P = \dfrac{w}{t} = \dfrac{{VIt}}{t} = VI$
Therefore, the power is directly proportional to voltage. Thai is,
$P \propto V$

Hence, option B is the correct answer.

Additional information:
The energy dissipated in time $t$ is given by $w = VIt$. This is derived from Joule’s Law equation. According to this equation, $H = VIt$ where $H$ is the heat produced by the electric current, $V$ is the voltage, $I$ is the current and $t$ is the time.

Note: The electrical power dissipated is $P = VI$.According to Ohm’s Law, $V = IR$. Substituting this in the above equation.
$P = \left( {IR} \right)I = {I^2}R$
Then, if we substitute$I = \dfrac{V}{R}$. Therefore,

\Rightarrow P = {\left( {\dfrac{V}{R}} \right)^2}R \\\ \Rightarrow P= \dfrac{{{V^2}}}{{{R^2}}} \times R\\\ \Rightarrow P = \dfrac{{{V^2}}}{R}$$ Therefore, power can also be written as $$P = \dfrac{{{V^2}}}{R}$$ Thus if one goes by this formula, the power is directly proportional to the square of voltage. That is,$$P \propto {V^2}$$. The power is said to be 1 Watt when a current of 1 Ampere is set up in the conductor across which a potential difference of 1 Volt is applied.