Question

Two electric bulbs rated \[{{P}_{1}}\] watts, $V$ volts and \[{{P}_{2}}\] watts, $V$ volt are connected in parallel and applied across $V$ volts. The total power (in watt) will be?
A. \[{{P}_{1}}+{{P}_{2}}\]
B. \[\sqrt{{{P}_{1}}{{P}_{2}}}\]
C. \[\dfrac{{{P}_{1}}{{P}_{2}}}{{{P}_{1}}+{{P}_{2}}}\]
D. \[\dfrac{{{P}_{1}}+{{P}_{2}}}{{{P}_{1}}{{P}_{2}}}\]

Answer 1

Hint:For the two electric bulbs, we know about their power and the voltage. We know power is given by \[P=\dfrac{{{V}^{2}}}{R}\] where V is the voltage across the resistor and R is its resistance. So, first of all we need to find out the resistances of both the resistance and they are connected in parallel, so we find out the equivalent resistance.

Complete step by step answer:
Resistance of first bulb: \[{{R}_{1}}=\dfrac{{{V}^{2}}}{{{P}_{1}}}\]
Resistance of second bulb: \[{{R}_{2}}=\dfrac{{{V}^{2}}}{{{P}_{2}}}\]
They are in parallel combination, so their equivalent resistance can be given as:
$R=\dfrac{\dfrac{{{V}^{2}}}{{{P}_{1}}}\times \dfrac{{{V}^{2}}}{{{P}_{2}}}}{\dfrac{{{V}^{2}}}{{{P}_{1}}}+\dfrac{{{V}^{2}}}{{{P}_{2}}}} \\
\Rightarrow R=\dfrac{{{V}^{4}}\times \dfrac{1}{{{P}_{1}}{{P}_{2}}}}{{{V}^{2}}[\dfrac{1}{{{P}_{1}}}+\dfrac{1}{{{P}_{2}}}]} \\
\Rightarrow R=\dfrac{{{V}^{2}}\times \dfrac{1}{{{P}_{1}}{{P}_{2}}}}{[\dfrac{1}{{{P}_{1}}}+\dfrac{1}{{{P}_{2}}}]} \\
\Rightarrow R=\dfrac{{{V}^{2}}\times \dfrac{1}{{{P}_{1}}{{P}_{2}}}}{\dfrac{{{P}_{1}}+{{P}_{2}}}{{{P}_{1}}{{P}_{2}}}} \\
\therefore R=\dfrac{{{V}^{2}}}{{{P}_{1}}+{{P}_{2}}} \\$
Now power, $P$= \[\dfrac{{{V}^{2}}}{R}\]
$\Rightarrow P= \dfrac{{{V}^{2}}}{\dfrac{{{V}^{2}}}{{{P}_{1}}+{{P}_{2}}}} \\
\therefore P={{P}_{1}}+{{P}_{2}} $

So, the correct option is A.

Additional Information:
In physics, an electric power measure of the rate of electrical energy transfer by an electric circuit per unit time. Denoted by P and measured using the SI unit of power is the watt or one joule per second. Electric power is commonly supplied by sources such as electric batteries and produced by electric generators.

Note:Electric power is used to find out how much current is consumed by that appliance when it is connected to power supply. The standard unit of measuring electric power is Watt, W. Another common unit is kilowatt kW. Power is defined as the ratio of work done to the time taken. Its unit is J/s or watt. $P=Fv$, in this equation we calculate the power is the instantaneous power.