Question

Write down the expression of equivalent resistance for ‘n’- number of resistors in parallel connection.

Answer 1

When resistors are connected in parallel, the total or equivalent resistance of the combination is different from when they are connected in series. For resistors in parallel, the reciprocal of the equivalent resistance \( R_{eq} \) is the sum of the reciprocals of each individual resistor's resistance.

If you have \( n \) resistors connected in parallel, with resistances \( R_1, R_2, \ldots, R_n \), the formula for the equivalent resistance \( R_{eq} \) is given by:

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots + \frac{1}{R_n} \]

To find the equivalent resistance \( R_{eq} \), you take the reciprocal of the sum:

\[ R_{eq} = \frac{1}{\left(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots + \frac{1}{R_n}\right)} \]

Here’s a step-by-step breakdown of the process:

1. **Find the Reciprocal of Each Resistance**: Calculate \( \frac{1}{R_i} \) for each resistor \( R_i \) where \( i \) ranges from 1 to \( n \).

2. **Sum the Reciprocals**: Add these reciprocals together: \( \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n} \).

3. **Take the Reciprocal of the Sum**: Finally, take the reciprocal of this sum to find the equivalent resistance \( R_{eq} \).

### Example

Suppose you have three resistors in parallel with resistances of \( 4 \,\Omega \), \( 6 \,\Omega \), and \( 12 \,\Omega \). The equivalent resistance is calculated as follows:

1. Compute the reciprocals:
   \[
   \frac{1}{R_1} = \frac{1}{4}, \quad \frac{1}{R_2} = \frac{1}{6}, \quad \frac{1}{R_3} = \frac{1}{12}
   \]

2. Sum the reciprocals:
   \[
   \frac{1}{4} + \frac{1}{6} + \frac{1}{12} = 0.25 + 0.1667 + 0.0833 = 0.5
   \]

3. Take the reciprocal of the sum:
   \[
   R_{eq} = \frac{1}{0.5} = 2 \,\Omega
   \]

So, the equivalent resistance of the three resistors in parallel is \( 2 \,\Omega \).

Answer 2

The equivalent resistance \( R_{eq} \) for \( N \)-number of resistors \( R_1, R_2, R_3, \dots, R_N \) in parallel connection is given by the reciprocal of the sum of the reciprocals of each individual resistance:

\[
\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_N}
\]

Or more generally:

\[
\frac{1}{R_{eq}} = \sum_{i=1}^{N} \frac{1}{R_i}
\]

Where \( R_{eq} \) is the equivalent resistance of the parallel combination of \( N \) resistors.