Equivalent Resistance of Identical Resistors in Parallel

When resistors are connected in parallel, the reciprocal of the equivalent resistance is equal to the sum of the reciprocals of individual resistances. This arrangement provides multiple paths for the current to flow.

The general formula for the equivalent resistance (\(R_{eq}\)) of resistors connected in parallel is:

\(\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}\)

In this specific problem, we are given that there are \(n\) identical resistors, and each resistor has a resistance of \(R\). So, \(R_1 = R_2 = ... = R_n = R\).

Substituting \(R\) for each individual resistance in the parallel formula, we get:

\(\frac{1}{R_{eq}} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} + ... + \frac{1}{R}\) (\(n\) times)

Adding the \(n\) identical terms:

\(\frac{1}{R_{eq}} = n \times \frac{1}{R}\)

\(\frac{1}{R_{eq}} = \frac{n}{R}\)

To find the equivalent resistance \(R_{eq}\), we take the reciprocal of both sides of the equation:

\(R_{eq} = \frac{R}{n}\)

Therefore, when \(n\) identical resistors, each of resistance \(R\), are connected in parallel, the equivalent resistance is \(\frac{R}{n}\).

Let's consider an example to solidify this concept. Suppose we have 4 identical resistors, each with a resistance of 10 ohms, connected in parallel. Here, \(n=4\) and \(R=10 \, \Omega\).

Using the formula \(R_{eq} = \frac{R}{n}\):

\(R_{eq} = \frac{10 \, \Omega}{4}\)

\(R_{eq} = 2.5 \, \Omega\)

This example shows how the equivalent resistance in a parallel connection of identical resistors is the individual resistance divided by the number of resistors.

Comparing this result with the given options:

• Option 1: \(n^2R\)
• Option 2: \(\frac{R}{n}\)
• Option 3: \(nR\) (This is for series connection)
• Option 4: \(\frac{R}{n^2}\)

The calculated equivalent resistance matches option 2.