Question

Calculate the equivalent resistance of three resistors \( R_1, R_2, R_3 \) connected in parallel.

Hint:

For resistors in parallel: - Voltage is same across each resistor, - Current divides, - Equivalent resistance is always less than the smallest resistor.

Answer 1

Concept: When resistors are connected in parallel, the potential difference across each resistor is the same, but the current divides among the branches. The total current is equal to the sum of currents through each resistor. Using Ohm’s Law: \[ I = \frac{V}{R} \] Derivation of Equivalent Resistance: Let the equivalent resistance of the parallel combination be \( R_{\text{eq}} \). If a potential difference \( V \) is applied across the combination: Current through each resistor: \[ I_1 = \frac{V}{R_1}, \quad I_2 = \frac{V}{R_2}, \quad I_3 = \frac{V}{R_3} \] Total current in the circuit: \[ I = I_1 + I_2 + I_3 \] \[ I = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3} \] But from Ohm’s Law for the equivalent resistance: \[ I = \frac{V}{R_{\text{eq}}} \] Equating both expressions for total current: \[ \frac{V}{R_{\text{eq}}} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3} \] Dividing both sides by \( V \): \[ \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \] Final Result: \[ R_{\text{eq}} = \left( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \right)^{-1} \] Thus, the reciprocal of the equivalent resistance is equal to the sum of reciprocals of individual resistances.