Question

All wires have the same resistance and equivalent resistance between A and B is ${R_y}$. Now keys are closed, then the equivalent resistance will become

A. \[\frac{{7{R_0}}}{3}\]
B. \[\frac{{7{R_0}}}{9}\]
C. \[7{R_0}\]
D. \[\frac{{{R_0}}}{3}\]

Answer 1

Hint: We understand how to solve resistor series and parallel combinations in any circuit. However, in this case, we have resistors in both parallel and series configurations. To solve this type of problem, we must divide it into series components and solve it repeatedly. The resistance in these circuits is estimated by resolving them piece by piece.



Formula used:
Parallel resistance’s equivalent resistance formula:
\[\frac{1}{{{{\rm{R}}_{{\rm{eq }}}}}} = \frac{1}{{{{\rm{R}}_{{\rm{1 }}}}}} + \frac{1}{{{{\rm{R}}_{{\rm{2 }}}}}} + \frac{1}{{{{\rm{R}}_{{\rm{3 }}}}}} + .....\]



Complete answer:
We have been provided in the given data that,
All wires have the same resistance and equivalent resistance between A and B is Ry.
Let us restructure the given circuit by breaking the circuit to simpler circuits for easy understanding, we have


Now, we have to calculate the equivalent resistance, for that we can write as
\[\frac{{{\rm{7}}{{\rm{R}}_{{\rm{0 }}}}}}{3} = {{\rm{R}}_{{\rm{eq }}}}\]
As, from the given circuit it is understood that,
\[{{\rm{R}}_{{\rm{0 }}}} = \frac{{\rm{R}}}{3}\]
Now, we have substitute the value obtained before, we get
\[{{\rm{R}}_{{\rm{eq }}}} = \frac{{7{\rm{R}}}}{9}\]
Therefore, when keys are closed, then the equivalent resistance will become \[{{\rm{R}}_{{\rm{eq }}}} = \frac{{7{\rm{R}}}}{9}\]
Hence, option B is the correct answer




Note:All wires have the same resistance and equivalent resistance between A and B is \[Ry\], then when keys are closed, the equivalent resistance will become \[\frac{{7{R_0}}}{9}\].
To generate the same impact on the circuit by allowing the same amount of current, one resistor can be used in place of two or more. This is referred to as efficient resistance.