Question

Three resistors of $ 2\, \Omega $, $ 3\, \Omega $, and $ 6\, \Omega $ are connected in parallel. What is the equivalent resistance of the combination?

• A. \( 1\, \Omega \)
• B. \( 2\, \Omega \)
• C. \( 3\, \Omega \)
• D. \( 4\, \Omega \)

Hint:

Key Fact: In parallel, \( \frac{1}{R_{\text{eq}}} = \sum \frac{1}{R_i} \)

Answer 1

The Correct Option is A

To find the equivalent resistance \( R_{\text{eq}} \) of resistors connected in parallel, we use the formula: 

\[\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\]

where \( R_1 = 2\, \Omega \), \( R_2 = 3\, \Omega \), and \( R_3 = 6\, \Omega \).

Substituting the given values into the formula:

\[\frac{1}{R_{\text{eq}}} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6}\]

Finding a common denominator (6) for the fractions:

\[\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}, \quad \frac{1}{6} = \frac{1}{6}\]

Adding the fractions:

\[\frac{1}{R_{\text{eq}}} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1\]

Thus:

\[R_{\text{eq}} = \frac{1}{1} = 1\, \Omega\]

Therefore, the equivalent resistance of the combination is \( 1\, \Omega \).