The maximum and the minimum equivalent resistance obtained by combining n identical resistors of resistance R, are `R_("max") and R_("min")` respectively. The ratio `(R_("max"))/(R_("min"))` is equal to

Options

A. `n`
B. `n^(2)`
C. `n^(2)-1`
D. `n^(3)`

Answer

To solve the problem of finding the ratio of the maximum equivalent resistance to the minimum equivalent resistance obtained by combining \( n \) identical resistors of resistance \( R \), we can follow these steps:

### Step 1: Calculate Maximum Equivalent Resistance
The maximum equivalent resistance occurs when all resistors are connected in series. The formula for total resistance \( R_{\text{max}} \) in series is:
\[ R_{\text{max}} = R + R + R + \ldots + R \quad (n \text{ times}) = nR \]

### Step 2: Calculate Minimum Equivalent Resistance
The minimum equivalent resistance occurs when all resistors are connected in parallel. The formula for total resistance \( R_{\text{min}} \) in parallel is:
\[ \frac{1}{R_{\text{min}}} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} + \ldots + \frac{1}{R} \quad (n \text{ times}) = \frac{n}{R} \]
Thus, we can rearrange this to find \( R_{\text{min}} \):
\[ R_{\text{min}} = \frac{R}{n} \]

### Step 3: Calculate the Ratio
Now, we need to find the ratio of the maximum resistance to the minimum resistance:
\[ \frac{R_{\text{max}}}{R_{\text{min}}} = \frac{nR}{\frac{R}{n}} = \frac{nR \cdot n}{R} = n^2 \]

### Final Answer
The ratio \( \frac{R_{\text{max}}}{R_{\text{min}}} \) is equal to:
\[ \boxed{n^2} \]

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