Resistance of a System of Resistors - Resistors in Parallel

Resistors are said to be connected in parallel when their ends are connected at both sides, as shown in figure. The figure shows three resistors R₁​, R₂​, and R₃​ connected in parallel between points $C$ and $D$. $V$ is the potential difference between $C$ and $D$.

Resistors in parallel

Let I₁​, I_2​, and I₃​ be the currents flowing through R₁​, R₂​, and R_3​ respectively. Then, the total current flowing through the circuit is:

I=I₁+I₂+I₃

Let Rp​ (p for parallel) be the effective resistance between $C$ and $D$. According to Ohm’s law:

I=`(V)/(R_P)`

Similarly,

I₁=`(V)/(R_1)`, I₂=`(V)/(R_2)`, I₃=`(V)/(R_3)`

Substituting these in equation (1) 

`(V)/(R_P)`=`(V)/(R_1)`+`(V)/(R_2)`+`(V)/(R_3)`

`(1)/(R_P)`=`(1)/(R_1)`+`(1)/(R_2)`+`(1)/(R_3)`

If n resistors are connected in parallel,

`(1)/(R_P)`=`(1)/(R_1)`+`(1)/(R_2)`+`(1)/(R_3)`+..........+`(1)/(R_n)`

Even if any one of the several bulbs connected in parallel becomes non-functional because of some damage to its filament, the circuit does not break as the current flows through the other paths, and the rest of the bulbs light up. When several bulbs are connected in parallel, they emit the same amount of light as when they are connected individually in the circuit, while bulbs connected in series emit less light than when connected individually.

1. The reciprocal of the effective resistance (R_p​) for resistors connected in parallel is equal to the sum of the reciprocals of their individual resistances, expressed as:

`(1)/(R_P)`=`(1)/(R_1)`+`(1)/(R_2)`+`(1)/(R_3)`+..........+`(1)/(R_n)`

2. The current through each resistor in a parallel connection is inversely proportional to its resistance. The total current in the circuit is the sum of the currents flowing through all individual resistors.

3. The potential difference across all resistors in a parallel arrangement is the same, as all resistors share the same voltage source.

4. The effective resistance in a parallel connection is always smaller than the resistance of the smallest individual resistor.

5. This configuration is commonly used to decrease the overall resistance in a circuit, enabling higher current flow.

• In a parallel combination, the potential difference across each resistor is the same as that across the terminals of the battery.
• The total current in a parallel circuit is equal to the sum of the currents in the individual branches, and the equivalent resistance is less than the smallest resistance connected.