When it comes to analyzing electrical circuits, one common task is to determine the equivalent resistance between two points. This is particularly useful in situations where multiple resistors are connected in various configurations, such as series or parallel. By finding the equivalent resistance, we can simplify the circuit and make further calculations or analysis easier. In this article, we will explore different methods and techniques to find the equivalent resistance between points A and B, providing valuable insights and examples along the way.

## Understanding Resistance

Before diving into finding the equivalent resistance, let’s first understand what resistance is. Resistance is a fundamental property of any material that opposes the flow of electric current. It is measured in ohms (Ω) and denoted by the symbol R. The higher the resistance, the more difficult it is for current to flow through a material.

Resistance can be influenced by various factors, including the length and cross-sectional area of a conductor, as well as the material’s resistivity. Different materials have different resistivities, which determine their inherent resistance. For example, copper has a lower resistivity compared to steel, making it a better conductor of electricity.

## Series and Parallel Resistors

When resistors are connected in a circuit, they can be arranged in two common configurations: series and parallel.

### Series Resistors

In a series configuration, resistors are connected end-to-end, forming a single path for current to flow through. The total resistance in a series circuit is the sum of the individual resistances. Mathematically, we can express this as:

R_{eq} = R_{1} + R_{2} + R_{3} + … + R_{n}

For example, let’s consider a series circuit with three resistors: R_{1} = 10Ω, R_{2} = 20Ω, and R_{3} = 30Ω. The equivalent resistance can be calculated as:

R_{eq} = 10Ω + 20Ω + 30Ω = 60Ω

### Parallel Resistors

In a parallel configuration, resistors are connected side by side, providing multiple paths for current to flow through. The total resistance in a parallel circuit can be calculated using the following formula:

1/R_{eq} = 1/R_{1} + 1/R_{2} + 1/R_{3} + … + 1/R_{n}

Let’s consider a parallel circuit with three resistors: R_{1} = 10Ω, R_{2} = 20Ω, and R_{3} = 30Ω. The equivalent resistance can be calculated as:

1/R_{eq} = 1/10Ω + 1/20Ω + 1/30Ω

1/R_{eq} = 3/30Ω + 2/30Ω + 1/30Ω

1/R_{eq} = 6/30Ω

1/R_{eq} = 1/5Ω

R_{eq} = 5Ω

## Combining Series and Parallel Resistors

In real-world circuits, it is common to have a combination of series and parallel resistors. To find the equivalent resistance in such cases, we need to apply a step-by-step approach.

### Step 1: Simplify Series Resistors

If there are any series resistors, we can simplify them by replacing them with a single resistor whose resistance is equal to the sum of the individual resistances. This simplification reduces the circuit to an equivalent circuit with fewer components.

### Step 2: Simplify Parallel Resistors

If there are any parallel resistors, we can simplify them by replacing them with a single resistor whose resistance is equal to the reciprocal of the sum of the reciprocals of the individual resistances. This simplification also reduces the circuit to an equivalent circuit with fewer components.

### Step 3: Repeat Steps 1 and 2

We repeat steps 1 and 2 until we have simplified the circuit to a point where there are no more series or parallel resistors to simplify. At this stage, we are left with the equivalent resistance between points A and B.

## Example Circuit

Let’s consider a practical example to illustrate the process of finding the equivalent resistance between points A and B. We have a circuit with four resistors arranged as shown below:

Using the step-by-step approach, let’s find the equivalent resistance:

### Step 1: Simplify Series Resistors

Resistors R_{1} and R_{2} are connected in series. Their equivalent resistance can be calculated as:

R_{eq1} = R_{1} + R_{2} = 10Ω + 20Ω = 30Ω

Replacing R_{1} and R_{2} with R_{eq1}, the circuit simplifies to:

### Step 2: Simplify Parallel Resistors

Resistors R_{eq1} and R_{3} are connected in parallel. Their equivalent resistance can be calculated as:

1/R_{eq2} = 1/R_{eq1} + 1/R_{3}

1/R_{eq2} = 1/30Ω + 1/30Ω

1/R_{eq2} = 2/30Ω

1/R_{eq2} = 1/15Ω

R_{eq2} = 15Ω

Replacing