Question

How do you find the Equivalent Resistance of a system?

Answer 1

Finding the equivalent resistance of a system depends on how the resistors are connected: either in series, parallel, or a combination of both. Let’s go through the different scenarios:

1. Resistors in Series:

When resistors are connected end-to-end, their total or equivalent resistance (\(R_{eq}\)) is the sum of all the individual resistances.

Formula:
\[
R_{eq} = R_1 + R_2 + R_3 + \dots
\]

Example:
If you have three resistors in series: \( R_1 = 2 \, \Omega, R_2 = 3 \, \Omega, R_3 = 5 \, \Omega \)
\[
R_{eq} = 2 + 3 + 5 = 10 \, \Omega
\]

2. Resistors in Parallel:

When resistors are connected side-by-side (parallel), the reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances.

Formula:
\[
\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots
\]
or
\[
R_{eq} = \left( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots \right)^{-1}
\]

Example:
If you have two resistors in parallel: \( R_1 = 6 \, \Omega, R_2 = 3 \, \Omega \)
\[
\frac{1}{R_{eq}} = \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6}
\]
\[
R_{eq} = \frac{6}{3} = 2 \, \Omega
\]

3. Combination of Series and Parallel:

In more complex circuits, you may have both series and parallel combinations. In this case:

1. Simplify the series or parallel sections step by step.
   

1. Start by combining the simplest series or parallel resistors and reduce the circuit gradually until you find the total resistance.
   

Example:

Consider this combination:

1. \( R_1 = 2 \, \Omega \) and \( R_2 = 3 \, \Omega \) are in series.
   

1. \( R_3 = 6 \, \Omega \) is in parallel with \( R_4 = 3 \, \Omega \).
   

1. First, find the equivalent resistance of \( R_3 \) and \( R_4 \) in parallel:
   

\[
\frac{1}{R_{eq2}} = \frac{1}{R_3} + \frac{1}{R_4} = \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6}
\]
\[
R_{eq2} = 2 \, \Omega
\]

1. Now, find the total resistance of \( R_1, R_2, \) and \( R_{eq2} \) in series:
   

\[
R_{total} = R_1 + R_2 + R_{eq2} = 2 + 3 + 2 = 7 \, \Omega
\]

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Key Tips:

1. Series: Add the resistances directly.
   

1. Parallel: Use the reciprocal method.
   

1. For combinations: Simplify step-by-step.
   

Would you like an example with more resistors, or is this explanation clear enough?