LEARNATHON

III

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Learn more### Theory:

**Parallel circuits:**

Components in parallel circuits are connected to the source in two or more loops. There are multiple paths for the electric charge to flow in a parallel circuit.

Even if the circuit is broken at any point in the loop, the current can flow through the circuit. Any electric appliances linked to the circuit will also function. Hence, the electrical wiring in our houses is made of parallel circuits.

Important!

The sum of the individual currents in each parallel branch in a parallel circuit equals the main current flowing into or out of the parallel branches. The potential difference across separate parallel branches are the same.

*Parallel connections of resistors*

Let the resistances of three resistors be \(R_1\), \(R_2\) and \(R_3\), connected parallel across point \(A\) and \(B\). Let '\(I\)' be the current flowing through the circuit.

The potential difference across the three resistances are the same and equal to the potential difference between point \(A\) and \(B\). This potential difference is measured using the voltmeter.

The current \(I\) starts from the positive terminal of the battery and reaches point \(A\). The current passing through the resistors \(R_1\), \(R_2\) and \(R_3\) is divided into three branches \(I_1\), \(I_2\) and \(I_3\), respectively.

According to Ohm's law, the current \(I_1\), \(I_2\) and \(I_3\) are given as,

$\begin{array}{l}{I}_{1}=\frac{V}{{R}_{1}}\\ \\ {I}_{2}=\frac{V}{{R}_{2}}\\ \\ {I}_{3}=\frac{V}{{R}_{3}}\end{array}$

Then, the total current passing through the circuit is

$I\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}{I}_{1}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}{I}_{2}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}{I}_{3}$

Substituting the values of \(I_1\), \(I_2\) and \(I_3\) in the above equation, we get

$I\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{V}{{R}_{1}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{V}{{R}_{2}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{V}{{R}_{3}}$ ---- (eq. 1)

Let \(R_P\) be the effective resistance of the parallel combination of resistors in the circuit. Then, the (eq. 1) becomes

$I\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{V}{{R}_{P}}$ ---- (eq. 2)

On combining (eq. 1) and (eq. 2), we get

$\begin{array}{l}\frac{V}{{R}_{P}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{V}{{R}_{1}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{V}{{R}_{2}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{V}{{R}_{3}}\\ \\ \frac{1}{{R}_{P}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{1}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{2}}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{1}{{R}_{3}}\end{array}$

Important!

Hence, the sum of the reciprocals of the individual resistances is equal to the reciprocal of the effective or equivalent resistance when several resistors are connected in parallel.

The effective or equivalent resistance is $\frac{R}{n}$ when the '\(n\)' number of resistors having an equal resistances '\(R\)' are connected in parallel.

The above statement can be given in the form of an equation as,

$\begin{array}{l}\frac{1}{{R}_{P}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{1}{R}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{1}{R}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{1}{R}\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}.....\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\frac{1}{R}\\ \\ \frac{1}{{R}_{P}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{n}{R}\\ \\ {R}_{P}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{R}{n}\end{array}$

Important!

The equivalent resistance in a parallel combination is less than the lowest of the individual resistances.