/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 164 The resistance of the series com... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 164

The resistance of the series combination of two resistances is \(S\). When they are joined in parallel, the total resistance is \(P\). 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 (C) 2.

Step by step solution

01

For two resistances \(R_1\) and \(R_2\) in series, the total series resistance is given by: \(S = R_1 + R_2\) For two resistances \(R_1\) and \(R_2\) in parallel, the total parallel resistance is given by: \(\frac{1}{P} = \frac{1}{R_1} + \frac{1}{R_2}\) We are given the relation between \(S\) and \(P\) as: \(S = n P\) #Step 2: Solve for one resistance in terms of the other and the total resistance#

First, let's solve the series resistance equation for \(R_1\): \(R_1 = S - R_2\) Now, let's substitute this expression into the parallel resistance equation: \(\frac{1}{P} = \frac{1}{S - R_2} + \frac{1}{R_2}\) #Step 3: Setup inequality and solve for R_2#
02

Since we are looking for the minimum value for \(n\), we need to find the smallest possible value of \(\frac{S}{P}\). Therefore, we will set an inequality with respect to \(R_2\): \(\frac{1}{P} > \frac{1}{S - R_2} + \frac{1}{R_2}\) Now, cross-multiply and simplify: \(R_2(S - R_2) > (S - R_2)R_2\) Notice that both sides are equal. Since we are looking for the smallest possible value of \(n\), we can take the equality case where: \(R_2(S - R_2) = (S - R_2)R_2\) #Step 4: Solve for n#

Now we have: \(S - 2R_2 = 0\) From the relation between \(S\) and \(P\): \(S = n P\) Substituting the value of \(S\) from the previous relation: \(n P - 2R_2 = 0\) Since the resistances are positive, in this case, the smallest possible value for \(n\) is when \(R_1 = R_2\). We can use the above equation to find the value of \(n\): \(n = 2\) So, the minimum possible value of \(n\) is: (C) 2

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