Several electric bulbs designed to be used on a $220\ V$ electric supply line, are rated $10\ W$. How many lamps can be connected in parallel with each other across the two wires of a $220\ V$ line if the maximum allowable current is $5\ A$?

Given: Two wires of a $220\ V$ line with the maximum allowable current $5\ A$. And a bulb designed to be used on a $220\ V$ electric supply line, is rated $10\ W$.

To do: 
To find the number of bulbs that can be connected in parallel with each other across the given two wires.

Solution:
 
Given,

Potential Difference, V = 220 V

The power rating of each bulb, P = 10 W

Total current, I = 5 A

Let the number of lamps in parallel be n.

First, we will find the resistance of each bulb.

We know that,

$P=V\times I\phantom{\rule{0ex}{0ex}}$

$P=V\times \frac{V}{R}$                   $(\because\ V=I\times R,\ I=\frac{V}{R})$
$P=\frac{{V}^{2}}{R}\phantom{\rule{0ex}{0ex}}$

$R=\frac{{V}^{2}}{P}\phantom{\rule{0ex}{0ex}}$

$R=\frac{(220{)}^{2}}{10}\phantom{\rule{0ex}{0ex}}$

$R=\frac{48400}{10}\phantom{\rule{0ex}{0ex}}$

$R=4840\Omega$

Resistance of 1 bulb = $4840\Omega$

For a flow of 5A current, we need to find the equivalent resistance of the circuit.

By Ohm's Law

$V=I\times {R}_{E}\phantom{\rule{0ex}{0ex}}$

${R}_{E}=\frac{V}{I}\phantom{\rule{0ex}{0ex}}$

${R}_{E}=\frac{220}{5}\phantom{\rule{0ex}{0ex}}$

${R}_{E}=44\Omega$

We know that, 

In parallel combination, the equivalent resistance is given by-
$\frac{1}{{\mathrm{R}}_{\mathrm{p}}}=\frac{1}{{\mathrm{R}}_{1}}+\frac{1}{{\mathrm{R}}_{2}}+\frac{1}{{\mathrm{R}}_{3}}+.......$

$\therefore \frac{1}{{\mathrm{R}}_{\mathrm{E}}}=\frac{1}{\mathrm{R}}+\frac{1}{\mathrm{R}}+\frac{1}{\mathrm{R}}+.......$ (n times)

$\frac{1}{{\mathrm{R}}_{\mathrm{E}}}=n\times \frac{1}{\mathrm{R}}\phantom{\rule{0ex}{0ex}}$

$\frac{1}{44}=n\times \frac{1}{4840}\phantom{\rule{0ex}{0ex}}$

$\therefore \mathrm{n}=\frac{4840}{44}\phantom{\rule{0ex}{0ex}}$

$\mathrm{n}=\frac{1210}{11}\phantom{\rule{0ex}{0ex}}$

$\mathrm{n}=110\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

So, 110 lamps can be connected in parallel.

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Updated on: 10-Oct-2022

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