How are resistances connected in parallel? What are their characteristics? Derive the formula for equivalent resistance.

Resistors in Parallel

When resistors are connected in parallel, they are arranged such that the opposite ends of each resistor are connected to the same two points in the circuit. In other words, they share the same potential difference (voltage) across them.

Characteristics of Resistors in Parallel:

1. Same Voltage: All resistors in parallel experience the same voltage across them.
2. Current Division: The total current entering the parallel network is divided among the individual resistors, with each resistor carrying a portion of the current.
3. Equivalent Resistance: The total or equivalent resistance for resistors in parallel is always less than the smallest individual resistance in the group.

Derivation of the Formula for Equivalent Resistance

To derive the formula for the equivalent resistance of resistors in parallel, we will use Ohm’s Law and the concept of current division.

Let’s assume we have n resistors connected in parallel. Their resistances are R1,R2,R3,…,Rn​.

1. Total Current: When a total voltage $V$ is applied across the parallel resistors, the total current I will split into individual currents I1,I2,I3…..In​ through each resistor according to Ohm’s Law.

   For each resistor, Ohm’s Law gives:

   I1=V/R1,I2=V/R2,I3=V/R3,…

2. Total Current: The total current $I$ is the sum of the individual currents:

   I=I1+I2+I3+⋯+In​

   Substituting Ohm’s Law for each individual current:

   I=V/R1+V/R2+V/R3+⋯+V/Rn​

   Factoring out $V$ from the right-hand side:

   I=V(1/R1+1/R2+1/R3+⋯+1/Rn)

3. Equivalent Resistance: By Ohm’s Law, the total current is also related to the equivalent resistance Req of the parallel combination:

   I=V/Req​

   Equating the two expressions for $I$:

   V/Req=V(1/R1+1/R2+1/R3+⋯+1/Rn)

   Simplifying and canceling $V$ from both sides:

   1/Req=1/R1+1/R2+1/R3+⋯+1/Rn​

Thus, the formula for the equivalent resistance Req of resistors in parallel is:

1/Req=∑i.1/Ri​

For Two Resistors:

For two resistors R1​ and R2 in parallel, the formula simplifies to:

1/Req=1/R1+1/R2​

or

Req=R1R2/R1+R2​​

Summary of Key Points:

• Parallel connection: Resistors are connected across the same voltage.
• Current division: The total current divides between the resistors.
• Formula: The equivalent resistance is found by summing the reciprocals of the individual resistances.