Question

What is the formula for Equivalent Resistance?

Answer 1

The equivalent resistance of a circuit depends on whether the resistors are arranged in series or parallel. Here’s a detailed breakdown of the formulas for both configurations:

### 1. Resistors in Series

When resistors are connected in series, the total or equivalent resistance (\(R_{eq}\)) is simply the sum of the individual resistances. This is because the current flowing through each resistor is the same, and the total voltage across the series combination is the sum of the voltages across each resistor.

**Formula:**
\[
R_{eq} = R_1 + R_2 + R_3 + \ldots + R_n
\]

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

### 2. Resistors in Parallel

When resistors are connected in parallel, the total or equivalent resistance (\(R_{eq}\)) is calculated using the reciprocal of the sum of the reciprocals of the individual resistances. In this configuration, the voltage across each resistor is the same, but the current can vary.

**Formula:**
\[
\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots + \frac{1}{R_n}
\]

To find \(R_{eq}\), you can take the reciprocal of the total obtained from the right side:
\[
R_{eq} = \frac{1}{\left( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots + \frac{1}{R_n} \right)}
\]

- **Example:** If you have three resistors in parallel: \(R_1 = 2 \, \Omega\), \(R_2 = 3 \, \Omega\), and \(R_3 = 6 \, \Omega\):
  \[
  \frac{1}{R_{eq}} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6}
  \]
  \[
  \frac{1}{R_{eq}} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1
  \]
  Therefore, \(R_{eq} = 1 \, \Omega\).

### Summary

- **Series:** \(R_{eq} = R_1 + R_2 + R_3 + \ldots\)
- **Parallel:** \(\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots\)

These formulas help you calculate the equivalent resistance in various circuit configurations, which is essential for analyzing electrical circuits.

Answer 2

The formula for equivalent resistance depends on whether the resistors are connected in series or in parallel. Here's a breakdown:

### 1. **Resistors in Series**

When resistors are connected end-to-end, their resistances add up. The formula for the equivalent resistance \( R_{eq} \) of resistors in series is:

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

where \( R_1, R_2, R_3, \ldots, R_n \) are the resistances of the individual resistors.

### 2. **Resistors in Parallel**

When resistors are connected such that both ends of each resistor are connected to the same two points, the reciprocal of the equivalent resistance \( R_{eq} \) is the sum of the reciprocals of the individual resistances. The formula for resistors in parallel is:

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

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

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

### **Mixed Series and Parallel**

For circuits with a combination of series and parallel resistors, you'll need to simplify the circuit step by step:

1. **Identify and combine resistors in series or parallel** to reduce the circuit to simpler forms.
2. **Repeat** the process until you have a single equivalent resistance for the whole circuit.

If you provide a specific circuit diagram, I can help with the detailed steps to find the equivalent resistance.