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Chapter 12: Problem 1677

The effective resistance of a n number of resistors connected in parallel in \(\mathrm{x} \mathrm{ohm}\). When one of the resistors is removed, the effective resistance becomes y ohm. The resistance of the resistor that is removed is.... (A) \(\\{(\mathrm{xy}) /(\mathrm{x}+\mathrm{y})\\}\) (B) \(\\{(\mathrm{xy}) /(\mathrm{y}-\mathrm{x})\\}\) (C) \((\mathrm{y}-\mathrm{x})\) (D) \(\sqrt{x y}\)

Short Answer

Expert verified
The resistance of the resistor that was removed is (B) \(\frac{xy}{y-x}\) ohm.

Step by step solution

01

Understand the Problem

There are n resistors connected in parallel with a total resistance of x ohms. If one resistor is removed, the total resistance becomes y ohms. The aim is to find the resistance of that removed resistor. This requires understanding the formula for total resistance in a parallel circuit.
02

Recall the Formula for Resistors in Parallel

The formula for n number of resistors (R1, R2, R3,..., Rn) connected in parallel can be written as: \[ \frac{1}{R} = \frac{1}{R1} + \frac{1}{R2} + \frac{1}{R3} + \ldots + \frac{1}{Rn} \] Where R is the total resistance. Now, before removing the resistor, let's say the resistance is R1 and the resistance of all other resistors in parallel is R2. So, R (total resistance before removing the resistor) is x and is calculated as: \[ \frac{1}{x} = \frac{1}{R1} + \frac{1}{R2} \] After removing the resistor, the resistance is y and is calculated as: \[ \frac{1}{y} = \frac{1}{R2} \]
03

Solve for R1

Now, to find R1, firstly, let's express \(\frac{1}{R1}\) in terms of x and y using the formulas from second step: \[ \frac{1}{R1} = \frac{1}{x} - \frac{1}{y} \] The value of resistor R1 (the resistor that has been removed) is just reciprocal of the above expression: \[ R1= \frac{1}{\frac{1}{R1}} \] This simplifies to: \[ R1= \frac{xy}{y-x} \]
04

Check for the Correct Option

Looking at the available options, we can see that option (B) corresponds to our calculated Result R1. Therefore, the resistance of the resistor that was removed is (B) \(\frac{xy}{y-x}\) ohm.

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Most popular questions from this chapter

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