Understanding Equivalent Resistance in Parallel Circuits

When resistors are connected in parallel, the electrical current has multiple paths to flow. The equivalent resistance of a parallel combination is the total resistance that a single resistor would need to have to produce the same overall effect on the circuit as the combined resistors.

The question asks us to find the equivalent resistance of three resistors with values 2 Ω, 4 Ω, and 8 Ω connected in parallel.

Formula for Parallel Resistance

For resistors connected in parallel, the reciprocal of the equivalent resistance (\(R_{eq}\)) is equal to the sum of the reciprocals of the individual resistances. If we have resistors \(R_1, R_2, R_3, \dots, R_n\) in parallel, the formula is:

\( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n} \)

Calculating the Equivalent Resistance

In this problem, we have three resistors with resistances \(R_1 = 2 \, \Omega\), \(R_2 = 4 \, \Omega\), and \(R_3 = 8 \, \Omega\). We will use the formula for parallel resistance:

\( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \)

Substitute the given values:

\( \frac{1}{R_{eq}} = \frac{1}{2 \, \Omega} + \frac{1}{4 \, \Omega} + \frac{1}{8 \, \Omega} \)

To add these fractions, we need to find a common denominator. The least common multiple of 2, 4, and 8 is 8.

• \( \frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8} \)
• \( \frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8} \)
• \( \frac{1}{8} = \frac{1}{8} \)

Now, add the fractions:

\( \frac{1}{R_{eq}} = \frac{4}{8} + \frac{2}{8} + \frac{1}{8} \)

\( \frac{1}{R_{eq}} = \frac{4 + 2 + 1}{8} \)

\( \frac{1}{R_{eq}} = \frac{7}{8 \, \Omega} \)

To find \(R_{eq}\), we take the reciprocal of both sides:

\( R_{eq} = \frac{8}{7} \, \Omega \)

Analyzing the Result

The equivalent resistance is \(R_{eq} = \frac{8}{7} \, \Omega\). Let's calculate the approximate value:

\( R_{eq} \approx 1.14 \, \Omega \)

Now, let's compare this value with the given options:

• Option 1: Less than 2 ohm (1.14 Ω is less than 2 Ω)
• Option 2: More than 2 ohm but less than 4 ohm (1.14 Ω is not in this range)
• Option 3: 4 ohm (1.14 Ω is not equal to 4 Ω)
• Option 4: 14 ohm (1.14 Ω is not equal to 14 Ω)

Our calculated equivalent resistance of \( \frac{8}{7} \, \Omega \approx 1.14 \, \Omega \) is less than 2 Ω. This matches Option 1.

Key Principle for Parallel Resistors

A crucial principle for parallel resistor combinations is that the equivalent resistance is always less than the smallest individual resistance in the combination. In this problem, the individual resistances are 2 Ω, 4 Ω, and 8 Ω. The smallest resistance is 2 Ω. Therefore, the equivalent resistance must be less than 2 Ω. Our calculated value, \( \frac{8}{7} \, \Omega \approx 1.14 \, \Omega \), is indeed less than 2 Ω, which is consistent with this principle.

Revision Table: Parallel Resistance Calculation

Additional Information: Resistors in Series vs. Parallel

It is helpful to understand the difference between series and parallel resistor connections and how their equivalent resistances are calculated.

• Series Connection: Resistors are connected end-to-end, forming a single path for current. The total equivalent resistance is the sum of the individual resistances.

  For \(R_1, R_2, \dots, R_n\) in series: \( R_{eq} = R_1 + R_2 + \dots + R_n \)

  In a series combination, 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 current. The reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances.

  For \(R_1, R_2, \dots, R_n\) in parallel: \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n} \)

  In a parallel combination, the equivalent resistance is always less than the smallest individual resistance.

Understanding these rules helps predict the range of the equivalent resistance in different circuit configurations.