Derive the expression for the equivalent resistance of three resistors \( R_1 \), \( R_2 \) and \( R_3 \) connected in parallel combination.
Show 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 \( R \), \( R_p = R/n \).
Step 1: Understand parallel combination.
In a parallel combination, all resistors are connected between the same two points. Therefore, the voltage across each resistor is the same, but the current divides among them.
Step 2: Draw a diagram for reference. Step 3: Let the total current and voltage.
Let a voltage \( V \) be applied across the parallel combination (between points A and B). Let the total current entering the combination be \( I \).
Step 4: Current divides among resistors.
The current \( I \) divides into three parts: \( I_1 \) through \( R_1 \), \( I_2 \) through \( R_2 \), and \( I_3 \) through \( R_3 \). By Kirchhoff's Current Law (KCL):
\[
I = I_1 + I_2 + I_3
\]
Step 5: Apply Ohm's law to each resistor.
Since the voltage across each resistor is the same \( V \):
\[
I_1 = \frac{V}{R_1}, \quad I_2 = \frac{V}{R_2}, \quad I_3 = \frac{V}{R_3}
\]
Step 6: Substitute into the current equation.
\[
I = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}
\]
\[
I = V \left( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \right)
\]
Step 7: Define equivalent resistance.
Let \( R_p \) be the equivalent resistance of the parallel combination. By Ohm's law, the total current \( I \) is also given by:
\[
I = \frac{V}{R_p}
\]
Step 8: Equate the two expressions for \( I \).
\[
\frac{V}{R_p} = V \left( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \right)
\]
Cancel \( V \) (since \( V \neq 0 \)):
\[
\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}
\]
Step 9: Final expression.
\[
\boxed{\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}}
\]
Where \( R_p \) is the equivalent resistance of the three resistors connected in parallel.
Step 10: Special case for two resistors.
For two resistors in parallel, the formula simplifies to:
\[
R_p = \frac{R_1 R_2}{R_1 + R_2}
\]
For three resistors, we can also write:
\[
R_p = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}}
\]
