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KCL and KVL

For the combi...

Question

For the combination of resistors shown in the following figure, find the equivalent resistance between $M$ and $N$

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Solution

Step 1: Given data
Step 2: Formula used
For resistors in parallel combination
$\frac{1}{R_{P}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + . . . + \frac{1}{R_{n}}$
Step 3: Finding the resultant resistances $R_{3}$ and $R_{4}$ which are connected in parallel
We know for resistors in parallel combination
$\frac{1}{R_{P}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + . . . + \frac{1}{R_{n}}$
Since $R_{3}$ and $R_{4}$ are parallel, their resultant,
$\frac{1}{R'} = \frac{1}{R_{3}} + \frac{1}{R_{4}} \Rightarrow R' = \frac{R_{3} R_{4}}{R_{3} + R_{4}}$
Step 4: Finding the resultant of all the four resistances $R_{1}, R_{2}, R_{3}$ and $R_{4}$
We know the resultant of resistances $R_{3}$ and $R_{4}$ is $R'$
Therefore the resultant of all four resistances $R_{1}, R_{2}, R_{3}$ and $R_{4}$ = The resultant of $R_{1}, R_{2}$ and $R'$
We know, for resistors in series combination:
$R_{S} = R_{1} + R_{2} + R_{3} + . . . + R_{n}$
Since $R_{1}, R_{2}$ and $R'$ are connected in series, their resultant,
$R = R_{1} + R_{2} + R' \Rightarrow R = R_{1} + R_{2} + \frac{R_{3} R_{4}}{R_{3} + R_{4}}$
Therefore, the equivalent resistance between $M$ and $N$ is $R_{1} + R_{2} + \frac{R_{3} R_{4}}{R_{3} + R_{4}}$ .

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