Calculating Equivalent Resistance in Series and Parallel Combinations

The question asks us to find the total equivalent resistance of a circuit arrangement involving both parallel and series connections of resistors. We have two identical resistors connected in parallel, and this parallel combination is then connected in series with a third resistor.

Let's break down the problem step-by-step:

• We have two identical resistors, each with a resistance of 10 Ω. Let's call them \(R_1\) and \(R_2\). So, \(R_1 = 10 \text{ Ω}\) and \(R_2 = 10 \text{ Ω}\).
• These two resistors (\(R_1\) and \(R_2\)) are connected in parallel.
• This parallel combination is then connected in series with a third resistor, \(R_3\), which also has a resistance of 10 Ω. So, \(R_3 = 10 \text{ Ω}\).

Step 1: Calculate the equivalent resistance of the parallel combination

When resistors are connected in parallel, the reciprocal of the equivalent resistance is the sum of the reciprocals of individual resistances. For two resistors \(R_1\) and \(R_2\) in parallel, the equivalent resistance \(R_p\) is given by the formula:

\(\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2}\)

Alternatively, for two resistors, a simpler formula is:

\(R_p = \frac{R_1 \times R_2}{R_1 + R_2}\)

Using the values \(R_1 = 10 \text{ Ω}\) and \(R_2 = 10 \text{ Ω}\):

\(R_p = \frac{10 \text{ Ω} \times 10 \text{ Ω}}{10 \text{ Ω} + 10 \text{ Ω}}\)

\(R_p = \frac{100 \text{ Ω}^2}{20 \text{ Ω}}\)

\(R_p = 5 \text{ Ω}\)

So, the equivalent resistance of the two 10 Ω resistors in parallel is 5 Ω.

Step 2: Calculate the total equivalent resistance of the combination

The parallel combination (with equivalent resistance \(R_p = 5 \text{ Ω}\)) is connected in series with the third resistor \(R_3 = 10 \text{ Ω}\). When resistors are connected in series, the total equivalent resistance is simply the sum of the individual resistances. Let \(R_{eq}\) be the total equivalent resistance.

\(R_{eq} = R_p + R_3\)

Substituting the values:

\(R_{eq} = 5 \text{ Ω} + 10 \text{ Ω}\)

\(R_{eq} = 15 \text{ Ω}\)

Therefore, the equivalent resistance of the entire combination is 15 Ω.

This step-by-step calculation shows how the equivalent resistance is determined for a circuit combining parallel and series resistor configurations.

Revision Table: Resistor Combinations

Additional Information on Equivalent Resistance

The concept of equivalent resistance simplifies circuit analysis. An equivalent resistor is a single resistor that can replace a network of resistors (series, parallel, or a combination) such that the total current and voltage relationships in the rest of the circuit remain unchanged.

• Series Connection: Resistors are connected end-to-end, forming a single path for the current. The same current flows through each resistor. The total voltage across the combination is the sum of the voltages across individual resistors. The equivalent resistance is always greater than the largest individual resistance.
• Parallel Connection: Resistors are connected across the same two points, providing multiple paths for the current. The voltage across each resistor is the same. The total current entering the junction is divided among the parallel paths. The equivalent resistance is always less than the smallest individual resistance.

Understanding how to calculate equivalent resistance for different configurations is fundamental in analyzing electrical circuits and applying Ohm's law.