Question

Find the equivalent resistance a cross \(\mathrm{AB}\) (A) \(1 \Omega\) (B) \(2 \Omega\) (C) \(3 \Omega\) (D) \(4 \Omega\)

Short answer

The given exercise does not provide enough information about the values of the resistors or their connections in the circuit. Please provide a circuit diagram or more information about the circuit components to find the equivalent resistance across AB.

Step-by-step solution

Identify Circuit Connections

First, we need to identify the connections between resistors in the circuit. In this case, there are no clarifications about the components present in the circuit. We will assume there are resistors with equal values, connected either in series or parallel.

Series Connection Formula

The total resistance \(R_s\) for resistors connected in series can be calculated using the series connection formula: \[ R_s = R_1 + R_2 + R_3 + ... + R_n \] where \(R_1, R_2, R_3, ..., R_n\) are the values of resistors connected in series.

Parallel Connection Formula

The total resistance \(R_p\) for resistors connected in parallel can be calculated using the parallel connection formula: \[ \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n} \] where \(R_1, R_2, R_3, ..., R_n\) are the values of resistors connected in parallel.

Calculate Equivalent Resistance

To find the equivalent resistance across AB, we will use the series and parallel connection formulas as needed for the given circuit configuration. Unfortunately, the given exercise does not provide enough information about the values of the resistors or their connections in the circuit. Please provide a circuit diagram or more information about the circuit components to find the equivalent resistance across AB.

Key concepts

Understanding Series Connection

In a **series connection**, resistors are arranged such that the current flows through each resistor one after the other. This means that the same current travels through every resistor connected in this way. The principle of a series connection is simple yet fundamental in understanding how circuits work. To compute the total or equivalent resistance in a series, you simply add the resistance values of each individual resistor.

For example, if resistors with resistance values of 2 Ω, 3 Ω, and 5 Ω are connected in series, the equivalent resistance, denoted by \( R_s \), can be calculated as follows:

• Sum the resistances: \( R_s = 2 + 3 + 5 = 10 \,\Omega \)

Thus, the total resistance in a series connection will always be greater than any individual resistance present in the circuit. This is because every added resistor increases the path's total opposition to current flow.

Grasping Parallel Connection

A **parallel connection** differs significantly from a series connection in terms of how resistors are connected in the circuit. In a parallel arrangement, all resistors are connected across the same two points or nodes, providing multiple paths for the current to travel through. This allows different parts of the current to flow through each resistor separately.

Calculating equivalent resistance in parallel is slightly more intricate than in series. For a parallel circuit, the formula used is:

• \( \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots + \frac{1}{R_n} \)

After calculating the reciprocal sum, take the reciprocal of the result to find the equivalent resistance \( R_p \).

For instance, if you have parallel-connected resistors with resistances of 4 Ω and 8 Ω, the equivalent resistance is found by

• Calculating: \( \frac{1}{R_p} = \frac{1}{4} + \frac{1}{8} = \frac{1}{2.67} \approx 2.67 \,\Omega \)

The crucial aspect of parallel connections is that the total resistance is always less than the smallest individual resistor's value. This occurs because the circuit's overall resistance decreases as additional paths for current are made available.

Diving into Circuit Analysis

**Circuit analysis** involves assessing a circuit to understand how various components interact and to determine values like voltage, current, and resistance in different parts of the circuit. Basic circuit analysis usually starts with identifying whether resistors are connected in series or parallel.

Once identified, you can use the respective formulas to calculate the equivalent resistance of the simplified circuit. This requires accurate recognition and simplification of the circuit into identifiable series and parallel sections.

• For series connections: apply \( R_s = R_1 + R_2 + R_3 + \ldots + R_n \)
• For parallel connections: use \( \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots \)

By calculating equivalent resistances, you can simplify even the most complex circuits.

Knowing how to break down circuits into series and parallel parts is crucial. This not only helps in finding the total resistance but also aids in understanding how a circuit will behave under varied conditions. Comprehensive circuit analysis offers insights that are central to both theoretical understanding and practical applications in electronics and electrical engineering.