The resistance of the series combination of two resistances is S. When they are joined in parallel,

Q: Why do we use the AM-GM inequality here?

The AM-GM inequality is a powerful mathematical tool used to find the minimum or maximum value of expressions involving the sum and product of positive numbers. Here, we have the sum of a number and its reciprocal ($\frac{R_1}{R_2} + \frac{R_2}{R_1}$), and their product is a constant (1). AM-GM perfectly gives the minimum value of such a sum.

Q: Can we solve this using calculus instead of AM-GM?

Yes, absolutely! You can let $x = \frac{R_1}{R_2}$, so $n = x + \frac{1}{x} + 2$. To find the minimum, take the derivative with respect to $x$, set it to zero ($1 - \frac{1}{x^2} = 0$), which gives $x = 1$. Substituting $x = 1$ back gives $n = 4$.

Q: What does $n=4$ physically mean for the resistors?

It means that for any two resistors, their series combination will always have a resistance that is at least 4 times greater than their parallel combination. This minimum ratio of 4 is achieved only when the two resistors have the exact same value ($R_1 = R_2$).

Q: Is it possible for $n$ to be less than 4?

No, it is mathematically impossible for real, positive resistances. The sum of a positive number and its reciprocal is always greater than or equal to 2. Therefore, $n$ will always be greater than or equal to 4.

Q: What if there were three resistors instead of two?

The relationship becomes more complex. For three identical resistors, $S = 3R$ and $P = R/3$, so $S = 9P$. The minimum ratio $S/P$ for $k$ identical resistors is $k^2$.

Question

The resistance of the series combination of two resistances is $S$. When they are joined in parallel, the total resistance is $P$. If $S = nP$, then the minimum possible value of $n$ is

Options:

(A) 4

(B) 3

(C) 2

(D) 1

Sigma Percentile · Accepted Answer

The relationship between series and parallel combinations of resistors hides a beautiful mathematical secret. In this problem, we are asked to find the minimum possible ratio between the series resistance ($S$) and the parallel resistance ($P$) of two resistors. Let's dive into the math and uncover this classic result!

### Setting Up the Equations

Imagine we have two resistors, $R_1$ and $R_2$. 
When we connect them in series, their equivalent resistance $S$ is simply their sum:
$$S = R_1 + R_2$$

When we connect them in parallel, their equivalent resistance $P$ is given by the product over sum formula:
$$P = \frac{R_1 R_2}{R_1 + R_2}$$

The problem gives us a fascinating condition: $S = nP$. Our goal is to find the minimum possible value for this multiplier $n$.

### The Algebraic Manipulation

Let's substitute our expressions for $S$ and $P$ into the given condition:
$$R_1 + R_2 = n \left( \frac{R_1 R_2}{R_1 + R_2} ight)$$

To isolate $n$, we can cross-multiply by $(R_1 + R_2)$ and divide by $R_1 R_2$:
$$n = \frac{(R_1 + R_2)^2}{R_1 R_2}$$

Now, let's expand the numerator using the algebraic identity $(a+b)^2 = a^2 + b^2 + 2ab$:
$$n = \frac{R_1^2 + R_2^2 + 2R_1 R_2}{R_1 R_2}$$

By dividing each term in the numerator by the denominator $R_1 R_2$, we get a beautifully symmetric expression:
$$n = \frac{R_1}{R_2} + \frac{R_2}{R_1} + 2$$

### The Magic of AM-GM Inequality

Look closely at the first two terms: $\frac{R_1}{R_2}$ and $\frac{R_2}{R_1}$. They are exact reciprocals of each other! In mathematics, whenever we want to find the minimum value of the sum of a positive number and its reciprocal, the **Arithmetic Mean-Geometric Mean (AM-GM) inequality** is our best friend.

The AM-GM inequality states that for any two positive numbers $a$ and $b$:
$$\frac{a + b}{2} \ge \sqrt{ab}$$

Let's apply this to our reciprocal terms, setting $a = \frac{R_1}{R_2}$ and $b = \frac{R_2}{R_1}$:
$$\frac{\frac{R_1}{R_2} + \frac{R_2}{R_1}}{2} \ge \sqrt{\frac{R_1}{R_2} 	imes \frac{R_2}{R_1}}$$

Notice how the terms inside the square root cancel each other out perfectly, leaving just $\sqrt{1}$, which is $1$.
$$\frac{\frac{R_1}{R_2} + \frac{R_2}{R_1}}{2} \ge 1$$
$$\frac{R_1}{R_2} + \frac{R_2}{R_1} \ge 2$$

### The Final Result

Now that we know the minimum value of the reciprocal sum is $2$, we can substitute this back into our equation for $n$:
$$n_{min} = 2 + 2 = 4$$

**The minimum possible value of $n$ is 4.**

This is a profound physical result: **The series resistance of any two resistors is always at least four times their parallel resistance.** This minimum ratio of 4 is achieved only when the two resistors are perfectly identical ($R_1 = R_2$). Keep this golden rule in your toolkit; it's a fantastic shortcut for analyzing circuits!

Animated Solution for Physics - Current Electricity: 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

Select Answer:

Visualized Solution

$S$ and $P$ in terms of $R_{1}$ and $R_{2}$

Using the given condition $S = n P$

Expressing $n$ in terms of $R_{1}$ and $R_{2}$

Applying AM-GM Inequality

Finding the minimum value of $n$

Conclusion and Reflection

The Sigma Insight: Combination of Resistors

Setting Up the Equations

The Algebraic Manipulation

The Magic of AM-GM Inequality

The Final Result

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