Question

Derive the expression for the equivalent resistance of three resistors \( R_1 \), \( R_2 \) and \( R_3 \) connected in parallel combination.

Hint:

In parallel combination: Voltage same across all resistors. Current divides. Equivalent resistance: \( \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \). For n equal resistors \( R \), \( R_p = R/n \).

Answer 1

Derivation of Equivalent Resistance for Three Resistors in Parallel:

Consider three resistors \( R_1 \), \( R_2 \) and \( R_3 \) connected in parallel across a battery of voltage \( V \).

Step 1: Key Property of Parallel Combination
In parallel combination: – The potential difference across each resistor is the same.
– The total current is equal to the sum of currents flowing through each resistor. So, \[ V_1 = V_2 = V_3 = V \]

Step 2: Apply Ohm’s Law to Each Resistor
Using Ohm’s Law: \[ I = \frac{V}{R} \] Current through each resistor: \[ I_1 = \frac{V}{R_1} \] \[ I_2 = \frac{V}{R_2} \] \[ I_3 = \frac{V}{R_3} \]

Step 3: Total Current in Parallel Combination
Total current: \[ I = I_1 + I_2 + I_3 \] Substitute values: \[ I = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3} \] Factor out V: \[ I = V \left( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \right) \]

Step 4: Define Equivalent Resistance
Let equivalent resistance be \( R \). By Ohm’s law for the whole combination: \[ I = \frac{V}{R} \]

Step 5: Equate the Two Expressions for Current
\[ \frac{V}{R} = V \left( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \right) \] Cancel V from both sides: \[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \]

Final Expression: \[ \boxed{\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}} \]

Conclusion:
The reciprocal of the equivalent resistance of resistors connected in parallel is equal to the sum of the reciprocals of their individual resistances. This shows that the equivalent resistance in parallel is always less than the smallest individual resistance.