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Chapter 12: Problem 1812

The resistance of the series combination of two resistances is \(\mathrm{S}\), when they are joined in parallel the total resistance is \(\mathrm{P}\) If $S=n P\(, then the minimum possible value of \)n$ is.... (A) 4 (B) 3 (C) 2 (D) 1

Short Answer

Expert verified
The minimum possible value of \(n\) is 4. (A)

Step by step solution

01

Recall the formulas for calculating resistance in series and parallel

Let's denote the two resistances as \(R_1\) and \(R_2\). The formula for calculating the total resistance in series (\(S\)) and parallel (\(P\)) are as follows: For series: \(S = R_1 + R_2\) For parallel: \(\displaystyle P = \frac{R_{1} R_{2}}{R_{1} +R_{2}}\) We are given that \(S = nP\) and our task is to find the minimum value of \(n\).
02

Substitute the formulas for series and parallel into the equation

Replace the values of \(S\) and \(P\) with their respective formulas from Step 1 into the equation \(S = nP\): \(R_1 + R_2 = n \cdot \frac {R_1R_2}{R_1+R_2}\)
03

Simplify the equation and eliminate common terms

The equation from Step 2 can be simplified by multiplying both sides by \((R_1+R_2)\): \((R_1 + R_2)^2 = n R_1 R_2\) Now, expand the left side of the equation: \(R_1^2 + 2R_1R_2 + R_2^2 = n R_1 R_2\)
04

Minimize the value of n

To find the minimum value of \(n\), we will look for the minimum possible non-negative difference between the left side and right side of the equation. This occurs when the equation becomes an equality, which means the right term of the left side (\(R_1^2 + R_2^2\)) should be as small as possible. The smallest value for the sum of squares of two positive values (as resistances are always positive) occurs when both \(R_1\) and \(R_2\) are equal. In that case, \(R_1 = R_2\) Now, substitute this condition into the equation from Step 3: \((R_1 + R_1)^2 = n R_1 R_1\) \(4R_1^2 = n R_1^2\)
05

Solve for n

Now divide both sides by \(R_1^2\) to find the minimum possible value of n: \(n = \frac{4R_1^2}{R_1^2}\) \(n = 4\) So, the minimum possible value of \(n\) is 4, which corresponds to option (A).

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Most popular questions from this chapter

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) \(\left(1 / \mathrm{n}^{2}\right)\)

A wire is bent in the form of a circle of radius \(4 \mathrm{~m}\) Resistance per unit length of wire is \(1 / \pi \Omega / \mathrm{m}\) battery of $6 \mathrm{~V}\( is connected between \)\mathrm{A}\( and \)\mathrm{B} \angle \mathrm{AOB}=90^{\circ}$ Find the current through the battery (A) \(8 \mathrm{~A}\) (B) \(4 \mathrm{~A}\) (C) \(3 \mathrm{~A}\) (D) \(9 \mathrm{~A}\)

Two resistors when connected in parallel have an equivalent of \(2 \Omega\) and when in series of \(9 \Omega\) The values of the two resistors are. (A) \(2 \Omega\) and \(9 \Omega\) (B) \(3 \Omega\) and \(6 \Omega\) (C) \(4 \Omega\) and \(5 \Omega\) (D) \(2 \Omega\) and \(7 \Omega\)

Two wires of resistances \(\mathrm{R}_{1}\) and \(\mathrm{R}_{2}\) have temperature coefficient of resistances \(\alpha_{1}\) and \(\alpha_{2}\) respectively they are joined in series the effective temperature coefficient of resistance is .... (A) \(\left[\left(\alpha_{1}+\alpha_{2}\right) / 2\right]\) (B) \(\sqrt{\alpha_{1} \alpha_{2}}\) (C) $\left[\left(\alpha_{1} R_{1}+\alpha_{2} R_{2}\right) /\left(R_{1}+R_{2}\right)\right]$ (D) $\left[\sqrt{\left(R_{1}\right.} R_{2} \alpha_{1} \alpha_{2}\right) / \sqrt{ \left.\left(R_{1}^{2}+R_{2}^{2}\right)\right]}$

In the circuit shown in fig the potential difference across \(3 \Omega\) is. (A) \(2 \mathrm{~V}\) (B) \(4 \mathrm{~V}\) (C) \(8 \mathrm{~V}\) (D) \(16 \mathrm{~V}\)
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