Question

State the law of combination of resistances in parallel.

Answer 1

Hint: Resistors are said to be in a parallel state when both the terminals of the resistors are connected to each terminal of the other resistors. The resistors in parallel have a common voltage across all the resistors. Here, we will use four resistors connected in parallel to find the law of combination of resistances.

Complete answer:
The figure below shows the circuit in which the resistors are placed parallel to each other.

Let $R_1$ , $R_2$ , $R_3$ and $R_4$ are resistors that are connected parallel to each other. Here, the voltage drop across each resistor will be the same but the electric current in the circuit will divide itself to travel through all the different branches. Now. to derive the equation of resistance in parallel, we will use Ohm’s law which is given by
$V=IR$
$I={\displaystyle \frac{V}{R}}$
Here, $I$ is the current in the circuit, $V$ is the voltage, and $R$ is the resistance in the circuit.

Now, according to Kirchhoff’s law, we get
$\sum I_{in}=\sum I_{out}$
$\Rightarrow I=I_1+I_2$
$I={\displaystyle \frac{V_1}{R_1}}+{\displaystyle \frac{V_2}{R_2}}$
Since the voltage drop across the resistors is the same. Therefore,
$I={\displaystyle \frac{V}{R_1}}+{\displaystyle \frac{V}{R_2}}$
$\Rightarrow I=V\left({\displaystyle \frac{1}{R_1}}-{\displaystyle \frac{1}{R_2}}\right)$
$\Rightarrow {\displaystyle \frac{I}{V}}=\left({\displaystyle \frac{1}{R_1}}-{\displaystyle \frac{1}{R_2}}\right)$
$\Rightarrow R_p={\left({\displaystyle \frac{1}{R_1}}-{\displaystyle \frac{1}{R_2}}\right)}^{-1}$
Therefore, the resistance in the parallel series are
$\therefore R_p={\left({\displaystyle \frac{1}{R_1}}+{\displaystyle \frac{1}{R_2}}+......+{\displaystyle \frac{1}{R_{n-1}}}+{\displaystyle \frac{1}{R_n}}\right)}^{-1}$

This means that the combined resistance in the parallel circuit will be equal to the sum of the reciprocal of all the individual resistances.Therefore, the law of combination of resistances in parallel states that the reciprocal of the combined resistance of all the resistors connected in parallel is equal to the sum of the reciprocal of all the individual resistance.

Note:In a parallel resistor circuit, the voltage across all the resistors will be the same. Therefore, the voltage in the resistor $R_1$ will be equal to the voltage in the resistor $R_2$ and is also equal to the voltage in the resistor $R_3$ . That is why we have used the same voltage for all the resistances.