Question

The resistors of values \(3\Omega\), \(6\Omega\) and \(18\Omega\) are connected in parallel in a circuit. The equivalent resistance in the circuit is

• A. \(12\Omega\)
• B. \(27\Omega\)
• C. \(18\Omega\)
• D. \(1.8\Omega\)

Hint:

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

Answer 1

The Correct Option is D

Concept:
When resistors are connected in parallel, the reciprocal of the equivalent resistance is equal to the sum of reciprocals of individual resistances. The formula for equivalent resistance in parallel combination is: \[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \] where: \[ R_1 = 3\Omega, \quad R_2 = 6\Omega, \quad R_3 = 18\Omega \]

Step 1: Substituting the given resistor values into the formula.
\[ \frac{1}{R} = \frac{1}{3} + \frac{1}{6} + \frac{1}{18} \] Now we take the LCM of \(3\), \(6\), and \(18\), which is \(18\). \[ \frac{1}{R} = \frac{6}{18} + \frac{3}{18} + \frac{1}{18} \] \[ \frac{1}{R} = \frac{6+3+1}{18} \] \[ \frac{1}{R} = \frac{10}{18} \] \[ \frac{1}{R} = \frac{5}{9} \]

Step 2: Finding the equivalent resistance.
Taking reciprocal on both sides: \[ R = \frac{9}{5} \] \[ R = 1.8\Omega \] Therefore, the equivalent resistance of the parallel combination is: \[ \boxed{1.8\Omega} \] Hence, the correct option is: \[ \boxed{(4)\ 1.8\Omega} \]