A piece of conducting wire of resistance R is cut into 2n equal parts. Half the parts are connected in series to form a bundle and remaining half in parallel to form another bundle. These bundles are then connected to give the maximum resistance. The maximum resistance of the combination is

Options

A. `R/2(1+1/n^2)`
B. `R/2 (1+n^2)`
C. `R/(2(1+n^2))`
D. `R (n+1/n)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Determine the resistance of each piece of wire
The total resistance of the wire is \( R \) and it is cut into \( 2n \) equal parts. Therefore, the resistance of each piece \( R_p \) is given by:

\[ R_p = \frac{R}{2n} \]

### Step 2: Calculate the equivalent resistance of the series bundle
Half of the pieces, which is \( n \) pieces, are connected in series. The equivalent resistance \( R_s \) of \( n \) resistors in series is the sum of their resistances:

\[ R_s = n \cdot R_p = n \cdot \frac{R}{2n} = \frac{R}{2} \]

### Step 3: Calculate the equivalent resistance of the parallel bundle
The remaining \( n \) pieces are connected in parallel. The equivalent resistance \( R_p \) of \( n \) resistors in parallel is given by:

\[ \frac{1}{R_p} = \frac{1}{R_p} + \frac{1}{R_p} + \ldots + \frac{1}{R_p} \quad (n \text{ times}) \]

This simplifies to:

\[ \frac{1}{R_p} = n \cdot \frac{1}{R_p} \implies R_p = \frac{R_p}{n} = \frac{R/2n}{n} = \frac{R}{2n^2} \]

### Step 4: Combine the two bundles to find the total resistance
Now, we have two equivalent resistances: \( R_s = \frac{R}{2} \) from the series connection and \( R_p = \frac{R}{2n^2} \) from the parallel connection. To find the total resistance \( R_t \) when these two bundles are connected in series, we add them:

\[ R_t = R_s + R_p = \frac{R}{2} + \frac{R}{2n^2} \]

### Step 5: Simplify the total resistance
To simplify \( R_t \):

\[ R_t = \frac{R}{2} + \frac{R}{2n^2} = \frac{R}{2} \left(1 + \frac{1}{n^2}\right) \]

### Final Result
Thus, the maximum resistance of the combination is:

\[ R_t = \frac{R}{2} \left(1 + \frac{1}{n^2}\right) \]