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Chapter 12: Problem 1747

There are n resistors having equal value of resistance \(\mathrm{r}\). First they are connected in such a way that the possible minimum value of resistance is obtained. Then they are connected in such a way that possible maximum value of resistance is obtained the ratio of minimum and maximum values of resistances obtained in these way is.... (A) \((1 / n)\) (B) \(\mathrm{n}\) (C) \(\mathrm{n}^{2}\) (D) \(\left(1 / \mathrm{n}^{2}\right)\)

Short Answer

Expert verified
The ratio of minimum and maximum values of resistances obtained from the connection combinations of n resistors with equal resistance r is \(\frac{1}{n^{2}}\), which corresponds to option (D).

Step by step solution

01

Calculate the total resistance for the parallel connection combination

For parallel connections, the formula for total resistance is given by: \(\frac{1}{R_{total}} = \frac{1}{r_1} + \frac{1}{r_2} + \cdots + \frac{1}{r_n}\) Since all the resistors have the same resistance r, the formula becomes: \(\frac{1}{R_{parallel}} = \frac{1}{r} + \frac{1}{r} + \cdots + \frac{1}{r}\) (n times) which simplifies to: \(\frac{1}{R_{parallel}} = n\cdot \frac{1}{r}\) Therefore, the total resistance in parallel connection is: \(R_{parallel} = \frac{r}{n}\)
02

Calculate the total resistance for the series connection

For series connections, the total resistance is simply the sum of the individual resistances: \(R_{series} = r_1 + r_2 + \cdots + r_n\) Since all the resistors have equal resistance r, we can simplify this to: \(R_{series} = n \cdot r\)
03

Determine the ratio between the minimum and maximum resistances

Now, we can find the ratio between R_parallel (minimum resistance) and R_series (maximum resistance): \(\frac{R_{parallel}}{R_{series}} = \frac{\frac{r}{n}}{n \cdot r}\) Simplify the equation by cancelling out r from both numerator and denominator: \(\frac{R_{parallel}}{R_{series}} = \frac{1}{n^{2}}\) Hence, the ratio of minimum and maximum values of resistances for these resistors is \( \frac{1}{n^{2}}\), which corresponds to option (D).

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Most popular questions from this chapter

A wire in a circular shape has \(10 \Omega\) resistance. The resistance per one meter is \(1 \Omega\) The resultant between \(A \& B\) is equal to \(2.4 \Omega\), then the length of the chord \(\mathrm{AB}\) will be equal to (A) \(2.4\) (B) 4 (C) \(4.8\) (D) 6

The drift velocity of free electrons in a conductor is \(\mathrm{V}\), when a current. \(I\) is flowing in it If both the radius and current are doubled, then drift velocity will be. (A) \((\mathrm{V} / 4)\) (B) \((\mathrm{V} / 2)\) (C) \(4 \mathrm{~V}\) (D) \(2 \mathrm{~V}\)

Figure, shows a network of eight resistors numbered 1 To 8 , each equal to $2 \Omega\(, connected to a \)3 \mathrm{~V}$ battery of negligible internal resistance The current \(\mathrm{I}\) in the circuit is .... (A) \(0.25 \mathrm{~A}\) (B) \(0.5 \mathrm{~A}\) (C) \(0.75 \mathrm{~A}\) (D) \(1.0 \mathrm{~A}\)

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