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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

The resistivity of a potentiometer wire is $40 \times 10^{-8} \mathrm{ohm} \mathrm{m}\( and its area of cross-section is \)8 \times 10^{-6} \mathrm{~m}^{2}\(. If \)0.2$ amp current is flowing through the wire, the potential gradient will be. (A) \(10^{-2}\) Volt \(/ \mathrm{m}\) (B) \(10^{-1}\) Volt \(/ \mathrm{m}\) (C) \(3.2 \times 10^{-2}\) Volt \(/ \mathrm{m}\) (D) 1 Volt \(/ \mathrm{m}\)

when a cell is connected to a resistance \(R_{1}\) the rate at which heat is generated in it is the same as when the cell is connected to a resistance \(\mathrm{R}_{2}\left(<\mathrm{R}_{1}\right)\) the internal resistance of the cell is.... (A) \(\left(R_{1}-R_{2}\right)\) (B) \((1 / 2)\left(\mathrm{R}_{1}-\mathrm{R}_{2}\right)\) (C) $\left\\{\left(\mathrm{R}_{1} \mathrm{R}_{2}\right) /\left(\mathrm{R}_{1}+\mathrm{R}_{2}\right)\right.$ (D) \(\sqrt{R}_{1} R_{2}\)

If \(\sigma_{1}, \sigma_{2}\), and \(\sigma_{3}\) are the conductance's of three conductor then equivalent conductance when they are joined in series, will be. (A) \(\sigma_{1}+\sigma_{2}+\sigma_{3}\) (B) $\left(1 / \sigma_{1}\right)+\left(1 / \sigma_{2}\right)+\left(1 / \sigma_{3}\right)$ (C) $\left\\{\left(\sigma_{1} \sigma_{2} \sigma_{3}\right) /\left(\sigma_{1}+\sigma_{2}+\sigma_{3}\right)\right\\}$ (D) None of these.

Two wires of equal diameters of resistivity's \(\rho_{1}\) and \(\rho_{2}\) are joined in series. The equivalent resistivity of the combination is.... (A) $\left\\{\left(\rho_{1} \ell_{1}+\rho_{2} \ell_{2}\right) /\left(\ell_{1}+\ell_{2}\right)\right\\}$ (B) $\left\\{\left(\rho_{1} \ell_{2}+\rho_{2} \ell_{1}\right) /\left(\ell_{1}-\ell_{2}\right)\right\\}$ (C) $\left\\{\left(\rho_{1} \ell_{2}+\rho_{2} \ell_{1}\right) /\left(\ell_{1}+\ell_{2}\right)\right\\}$ (D) $\left\\{\left(\rho_{1} \ell_{1}+\rho_{2} \ell_{2}\right) /\left(\ell_{1}-\ell_{2}\right)\right\\}$

On applying an electric field of \(5 \times 10^{-8} \mathrm{Vm}^{-1}\) across a conductor, current density through it is \(2.5 \mathrm{Am}^{-2}\) The resistivity of the conductor is \(\ldots .\) (A) \(1 \times 10^{-8} \Omega \mathrm{m}\) (B) \(2 \times 10^{-8} \Omega \mathrm{m}\) (C) \(0.5 \times 10^{-8} \Omega \mathrm{m}\) (D) \(12.5 \times 10^{-8} \Omega \mathrm{m}\)
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