A wire of resistance R is cut into ‘ n ’ equal parts. These parts are then connected in parallel. The equivalent resistance of the combination will be

Options

A. `( r )/(n )`
B. `( r )/( n^(2))`
C. `rn `
D. `n^(2) r `

Answer

To find the equivalent resistance of a wire of resistance \( R \) that is cut into \( n \) equal parts and connected in parallel, we can follow these steps:

### Step 1: Determine the resistance of each part
When the wire of resistance \( R \) is cut into \( n \) equal parts, the resistance of each part can be calculated as:
\[ R_{\text{each}} = \frac{R}{n} \]

### Step 2: Set up the formula for equivalent resistance in parallel
For resistors connected in parallel, the formula for equivalent resistance \( R_{\text{eq}} \) is given by:
\[ \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n} \]
Since all \( n \) resistances are equal (each is \( R_{\text{each}} = \frac{R}{n} \)), we can rewrite the equation as:
\[ \frac{1}{R_{\text{eq}}} = n \cdot \frac{1}{R_{\text{each}}} \]

### Step 3: Substitute the value of \( R_{\text{each}} \)
Substituting \( R_{\text{each}} = \frac{R}{n} \) into the equation, we have:
\[ \frac{1}{R_{\text{eq}}} = n \cdot \frac{1}{\frac{R}{n}} = n \cdot \frac{n}{R} = \frac{n^2}{R} \]

### Step 4: Solve for \( R_{\text{eq}} \)
Now, taking the reciprocal to find \( R_{\text{eq}} \):
\[ R_{\text{eq}} = \frac{R}{n^2} \]

### Final Answer
Thus, the equivalent resistance of the combination is:
\[ R_{\text{eq}} = \frac{R}{n^2} \]

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