Resistors connected in parallel: If the numbers of resistance are connected between two common points, such that the potential difference across each resistance is the same, then the arrangement is called resistance in parallel.
Three resistances R1, R2 and R3 are connected in parallel between the points A and B. Let Rp be the effective resistance in the circuit.
A Cell E, Key K and the ammeters A are also connected with resistances.
Let the current passing through R1 be I1, R2 be I2, and R3 be I3 and that of R be I.
Derivation of Equivalent Resistance in a Parallel Circuit
The image provided outlines the mathematical derivation for calculating the total equivalent resistance ($R_p$) when three resistors ($R_1$, $R_2$, and $R_3$) are connected in parallel. Below is a detailed, step-by-step breakdown of the physics principles shown.
Step 1: The Current Rule in Parallel Circuits
In a parallel circuit, the total current flowing from the source divides into the different parallel branches. Therefore, the total current ($I$) is the sum of the currents in each individual branch.
$$I = I_1 + I_2 + I_3 \quad \text{--- eq no (1)}$$
Step 2: Applying Ohm's Law
According to Ohm's law, Current ($I$) is equal to Voltage ($V$) divided by Resistance ($R$). A fundamental characteristic of parallel circuits is that the voltage ($V$) remains constant across all branches.
Applying this to each resistor gives us their individual currents:
$$I_1 = \frac{V}{R_1}, \quad I_2 = \frac{V}{R_2}, \quad I_3 = \frac{V}{R_3}$$
For the whole circuit, utilizing the total equivalent resistance ($R_p$), the total current is:
$$I = \frac{V}{R_p} \quad \text{--- eq no (2)}$$
Step 3: Substitution
By substituting the Ohm's law relationships from equation (2) into the total current equation (1), we replace the $I$ variables with their $\frac{V}{R}$ equivalents.
$$\frac{V}{R_p} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}$$
Step 4: Factoring Out Common Variables
Because the voltage ($V$) is identical in every term on the right side of the equation, it can be factored out as a common multiplier.
$$\frac{V}{R_p} = V \left[ \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \right]$$
Step 5: Final Formula for Parallel Resistance
Finally, dividing both sides by the common voltage $V$ cancels it out completely. This results in the standard formula used to calculate the equivalent resistance of resistors arranged in parallel.
$$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$$
Conclusion: If the resistors are connected in parallel then:
- The sum of reciprocals of the individual resistance is equal to the reciprocal of equivalent resistance.
- The current in various resistors are inversely proportional to the resistances (higher is the resistance lower is the current through it). However the total current is the sum of the currents flowing in the different branches.
- The voltage (potential difference) across each resistor is same.
- The effective resistance of the parallel combination is less than the individual resistance in the combination.
- This combination is used to decrease resistance in the circuit.