Question

What is equivalent resistance? Draw the expression for effective resistance of two resistors connected in parallel.

Answer 1

Hint: We calculate equivalent resistance when there are combinations of resistances in a circuit. Using the concept of total voltage and total current we can calculate equivalent resistance in a circuit with the help of Ohm’s law.
Formula used:
For series combination, ${{R}_{S}}={{R}_{1}}+{{R}_{2}}+{{R}_{3}}+......+{{R}_{n}}$
For parallel combination, $\dfrac{1}{{{R}_{P}}}=\dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}}+\dfrac{1}{{{R}_{3}}}+......+\dfrac{1}{{{R}_{n}}}$

Complete step by step answer:
When the circuit is designed in different combinations, series or parallel, we have to calculate the aggregate resistance, this value is known as equivalent resistance. Electrical resistance represents the amount of energy required to move charge or current in an electrical circuit. The equivalent resistance in a circuit or network is the value of that single resistor that can replace the whole network in such a way that for a certain value of applied voltage, we will get the same value of current as we would get in the original network.
Two ways in which resistances can be connected in a circuit are:
Series combination – Resistances are connected end to end along the same line. The value of current through each resistance remains the same. Value of voltage drop is different for each resistance.

Equivalent resistance formula for resistances connected in series:
${{R}_{S}}={{R}_{1}}+{{R}_{2}}+{{R}_{3}}+......+{{R}_{n}}$
Parallel combination – In this combination ends of resistances are connected at the same point. Current through each resistance will be different. Voltage across all the resistances will remain the same.

Equivalent resistance formula for resistances connected in parallel:
$\dfrac{1}{{{R}_{P}}}=\dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}}+\dfrac{1}{{{R}_{3}}}+......+\dfrac{1}{{{R}_{n}}}$
Let us consider two resistors ${{R}_{1}}$and ${{R}_{2}}$ are connected in parallel. Let the voltage source be $V$ and current passing through the circuit as $I$.

Current passing through ${{R}_{1}}$$={{I}_{1}}$ and through ${{R}_{2}}$$={{I}_{2}}$
Total current in the circuit $I={{I}_{1}}+{{I}_{2}}$
Put,$I=\dfrac{V}{R}$, ${{I}_{1}}=\dfrac{V}{{{R}_{1}}}$ and ${{I}_{2}}=\dfrac{V}{{{R}_{2}}}$
$\begin{align}
  & \dfrac{V}{R}=\dfrac{V}{{{R}_{1}}}+\dfrac{V}{{{R}_{2}}} \\
 & \dfrac{1}{R}=\dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}} \\
\end{align}$
Hence, the effective resistance of two resistors connected in parallel is given by:
$\dfrac{1}{R}=\dfrac{1}{{{R}_{1}}}+\dfrac{1}{{{R}_{2}}}$

Note: Students should keep in mind the concept of series and parallel combination; how resistances are joined in each case. Also remember, how voltage and current varies in each case. In parallel combination, voltage across each resistance remains the same while current gets divided. In series combination, current through each resistance remains the same while the voltage gets divided.