Understanding Resistors in Parallel Circuits

When resistors are connected in parallel, the current divides among them, and the voltage across each resistor is the same. The total resistance of a parallel combination is always less than the smallest individual resistance.

The formula for calculating the total resistance (\(\small R_{total}\)) of resistors connected in parallel is given by the reciprocal of the sum of the reciprocals of individual resistances:

\(\small \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...\)

Solving the Unknown Resistance Problem

In this specific problem, we are given three resistors connected in parallel with resistances \(\small R_1 = 9 \, \Omega\), \(\small R_2 = 9 \, \Omega\), and \(\small R_3 = X \, \Omega\). The total resistance of this parallel combination is \(\small R_{total} = 3 \, \Omega\). We need to find the value of the unknown resistance \(\small X\).

We can use the formula for the total resistance of resistors in parallel:

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

Step-by-Step Calculation

Substitute the given values into the formula:

\(\small \frac{1}{3} = \frac{1}{9} + \frac{1}{9} + \frac{1}{X}\)

Combine the terms with known values:

\(\small \frac{1}{3} = \frac{2}{9} + \frac{1}{X}\)

To find \(\small \frac{1}{X}\), rearrange the equation:

\(\small \frac{1}{X} = \frac{1}{3} - \frac{2}{9}\)

To subtract the fractions on the right side, find a common denominator, which is 9:

\(\small \frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9}\)

So the equation becomes:

\(\small \frac{1}{X} = \frac{3}{9} - \frac{2}{9}\)

Subtract the fractions:

\(\small \frac{1}{X} = \frac{3 - 2}{9}\)

\(\small \frac{1}{X} = \frac{1}{9}\)

To find \(\small X\), take the reciprocal of both sides:

\(\small X = 9 \, \Omega\)

Verification of the Result

Let's check if the calculated value of \(\small X = 9 \, \Omega\) gives the correct total resistance when combined in parallel with two other \(\small 9 \, \Omega\) resistors:

\(\small \frac{1}{R_{total}} = \frac{1}{9} + \frac{1}{9} + \frac{1}{9}\)

\(\small \frac{1}{R_{total}} = \frac{1+1+1}{9}\)

\(\small \frac{1}{R_{total}} = \frac{3}{9}\)

\(\small \frac{1}{R_{total}} = \frac{1}{3}\)

Taking the reciprocal:

\(\small R_{total} = 3 \, \Omega\)

This matches the total resistance given in the problem, confirming our calculation is correct.

Conclusion on Unknown Resistance

The unknown resistance \(\small X\) in the parallel combination is \(\small 9 \, \Omega\). This means all three resistors in the parallel circuit have the same resistance value.

Revision Table: Parallel Resistors

Additional Information: Parallel Circuit Analysis

Understanding parallel circuits is fundamental in electronics. Key characteristics include:

• The total resistance is always less than the smallest individual resistance. This is because adding more parallel paths allows current to flow more easily, reducing the overall opposition to current.
• Parallel connections are commonly used in household wiring. This ensures that each appliance receives the full line voltage (e.g., 120V or 240V), and if one appliance is turned off or fails, the others continue to operate.
• Using Ohm's Law (\(\small V = I \times R\)), you can find the current through each individual resistor if you know the voltage across the parallel combination and the resistance of each resistor. The total current is the sum of these individual currents.