Name: Meter Bridge
Uploaded: 2019-07-19
Duration: 7 min 2 s
Description: Measurement, by comparison, is always accurate. The video talks about a metre bridge device that helps us find the unknown resistance by comparing it with a...

Question

Derive an expression for the effective resistance when three resistors are connected in parallel.

Accepted Answer

When the resistors are connected such that one end of each resistor is connected together at one point and the other end is connected together at another point such that same potential difference is applied across each resistor then the resistors are said to be connected in parallel as shown in the below diagram:

Here, the three resistors connected in parallel are $R_{1}$ , $R_{2}$ and $R_{3}$ and the current flowing through them are $I_{1}$ , $I_{2}$ and $I_{3}$ respectively.

The potential difference applied across them is $V$ .

In parallel combination, the current divides among the resistors, the sum of which will be equal to the total current $I$ flowing through the combination.

Thus, $I = I_{1} + I_{2} + I_{3}$ ........(a)

But from Ohm's law, we get,

$I_{1} = \frac{V}{R_{1}}$ , $I_{2} = \frac{V}{R_{2}}$ and $I_{3} = \frac{V}{R_{3}}$ ......(b)

Now, substituting (b) in (a) we get,

$I = \frac{V}{R_{1}} + \frac{V}{R_{2}} + \frac{V}{R_{3}} = V (\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}})$

$\Rightarrow \frac{I}{V} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}$

But, $\frac{I}{V} = \frac{1}{R_{p}}$ where $R_{p}$ is the equivalent resistance of the resistors connected in parallel.

Hence, $\frac{1}{R_{p}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}$ is the expression for the effective resistance when three resistors are connected in parallel.

Derive an expression for the effective resistance when three resistors are connected in parallel.

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