Question

What is the equivalent resistance between two points in a circuit?

Answer 1

The equivalent resistance between two points in a circuit is the total resistance that can replace the combination of resistors between those points without changing the overall behavior of the circuit. To find this equivalent resistance, you need to consider how the resistors are connected: in series, parallel, or a combination of both.

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

When resistors are connected in series, the total or equivalent resistance \( R_{eq} \) is the sum of the individual resistances:

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

For example, if you have three resistors in series with resistances of \( 4 \ \Omega \), \( 6 \ \Omega \), and \( 10 \ \Omega \):

\[ R_{eq} = 4 \ \Omega + 6 \ \Omega + 10 \ \Omega = 20 \ \Omega \]

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

When resistors are connected in parallel, the reciprocal of the equivalent resistance \( R_{eq} \) is the sum of the reciprocals of the individual resistances:

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

For example, if you have three resistors in parallel with resistances of \( 4 \ \Omega \), \( 6 \ \Omega \), and \( 12 \ \Omega \):

\[ \frac{1}{R_{eq}} = \frac{1}{4 \ \Omega} + \frac{1}{6 \ \Omega} + \frac{1}{12 \ \Omega} = \frac{3}{12} + \frac{2}{12} + \frac{1}{12} = \frac{6}{12} = \frac{1}{2} \]

Thus:

\[ R_{eq} = 2 \ \Omega \]

### 3. **Combination of Series and Parallel**

For circuits that have both series and parallel components, you’ll need to break down the circuit into simpler series or parallel parts, find the equivalent resistance for each part, and then combine them accordingly.

#### Example:

Consider a circuit where a \( 10 \ \Omega \) resistor is in series with a parallel combination of \( 20 \ \Omega \) and \( 30 \ \Omega \) resistors.

1. **Find the equivalent resistance of the parallel resistors:**

   \[ \frac{1}{R_{parallel}} = \frac{1}{20 \ \Omega} + \frac{1}{30 \ \Omega} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \]

   Thus:

   \[ R_{parallel} = 12 \ \Omega \]

2. **Add this result to the series resistor:**

   \[ R_{eq} = 10 \ \Omega + 12 \ \Omega = 22 \ \Omega \]

By following these methods, you can determine the equivalent resistance between any two points in a circuit.

Answer 2

The equivalent resistance between two points in a circuit is the total resistance that an external voltage source "sees" when connected across those two points. It represents the combined effect of all the resistances between these two points. Determining the equivalent resistance depends on how the resistors are connected in the circuit, either in **series**, **parallel**, or a **combination** of both.

### Types of Resistor Connections

1. **Series Connection:**
   - Resistors are connected end-to-end.
   - The current through each resistor is the same.
   - The total or equivalent resistance is the sum of all individual resistances.

   \[
   R_{\text{eq(series)}} = R_1 + R_2 + R_3 + \ldots + R_n
   \]

   For example, if three resistors \(R_1 = 2 \, \Omega\), \(R_2 = 3 \, \Omega\), and \(R_3 = 5 \, \Omega\) are in series, the equivalent resistance is:

   \[
   R_{\text{eq}} = 2 + 3 + 5 = 10 \, \Omega
   \]

2. **Parallel Connection:**
   - Resistors are connected such that each end of every resistor is connected to the same two points.
   - The voltage across each resistor is the same.
   - The reciprocal of the equivalent resistance is the sum of the reciprocals of all individual resistances.

   \[
   \frac{1}{R_{\text{eq(parallel)}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots + \frac{1}{R_n}
   \]

   For example, if three resistors \(R_1 = 2 \, \Omega\), \(R_2 = 3 \, \Omega\), and \(R_3 = 6 \, \Omega\) are in parallel, the equivalent resistance is:

   \[
   \frac{1}{R_{\text{eq}}} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = 1
   \]

   Therefore,

   \[
   R_{\text{eq}} = 1 \, \Omega
   \]

3. **Combination of Series and Parallel:**
   - Many practical circuits involve a combination of resistors in series and parallel.
   - To find the equivalent resistance in such cases, one can simplify the circuit step-by-step by reducing series and parallel sections iteratively.

### Steps to Calculate Equivalent Resistance

1. **Identify Series and Parallel Sections:** Determine which resistors are in series and which are in parallel.
2. **Simplify Step-by-Step:** Start from the simplest parts of the circuit and reduce them to their equivalent resistances.
3. **Repeat the Process:** Continue this process until you have a single equivalent resistance for the entire circuit.

### Example Problem

Consider a circuit where \(R_1 = 4 \, \Omega\) and \(R_2 = 6 \, \Omega\) are in series, and their combination is in parallel with \(R_3 = 3 \, \Omega\).

- First, find the equivalent resistance of the series resistors:

  \[
  R_{1+2} = R_1 + R_2 = 4 + 6 = 10 \, \Omega
  \]

- Now, this \(10 \, \Omega\) resistance is in parallel with \(R_3 = 3 \, \Omega\):

  \[
  \frac{1}{R_{\text{eq}}} = \frac{1}{R_{1+2}} + \frac{1}{R_3} = \frac{1}{10} + \frac{1}{3}
  \]

  \[
  \frac{1}{R_{\text{eq}}} = \frac{3}{30} + \frac{10}{30} = \frac{13}{30}
  \]

  \[
  R_{\text{eq}} = \frac{30}{13} \approx 2.31 \, \Omega
  \]

Therefore, the equivalent resistance between the two points in this circuit is approximately \(2.31 \, \Omega\).

### Conclusion

Calculating the equivalent resistance between two points in a circuit involves understanding the configuration of the resistors. The steps include identifying which resistors are in series or parallel and simplifying the circuit step-by-step. This concept is fundamental in circuit analysis and is widely used in electrical engineering and physics.