Question

What is the principle of parallel theorem?

Answer 1

The parallel theorem in electrical circuits is a rule used to simplify the analysis of resistors in parallel. The principle of the parallel theorem states that:

In a parallel circuit, the total or equivalent resistance (R_eq) is always less than the smallest resistance in the group.

This is based on the fact that in a parallel connection, multiple paths for current are provided, making it easier for current to flow. This results in a lower total resistance.

Key Points:

1. Formula for Parallel Resistance:  
   

   For two resistors \( R_1 \) and \( R_2 \) in parallel, the total resistance \( R_{\text{eq}} \) is given by:
   \[
   \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2}
   \]
   For three resistors \( R_1 \), \( R_2 \), and \( R_3 \), the formula becomes:
   \[
   \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}
   \]

1. More Resistors, Less Resistance: Adding more resistors in parallel will always decrease the total resistance, but the rate of decrease slows as more resistors are added.
   

1. Total Resistance is Always Lower: The total resistance for parallel resistors will always be lower than the smallest individual resistor. For example, if you have resistors of 5Ω and 10Ω in parallel, the equivalent resistance will be less than 5Ω.
   

Why It Works:

When resistors are connected in parallel, the current has multiple paths to take. Since current is shared across these paths, the overall opposition to current (resistance) is reduced.

Example:

If you have a 4Ω resistor and a 6Ω resistor in parallel, the total resistance is:
\[
\frac{1}{R_{\text{eq}}} = \frac{1}{4} + \frac{1}{6} = \frac{5}{12}
\]
So, \( R_{\text{eq}} = \frac{12}{5} = 2.4 \, \Omega \).  
The total resistance is 2.4Ω, which is lower than either 4Ω or 6Ω.