When a wire of uniform cross-section `a`, length `I` and resistance `R` is bent into a complete circle, resistance between two of diametrically opposite points will be

Options

A. `(R )/(4)`
B. `(R )/(8)`
C. `(4 R)`
D. `(R )/(2)`

Explanation

To find the resistance between two diametrically opposite points on a wire of uniform cross-section that has been bent into a complete circle, we can follow these steps:

### Step 1: Understand the initial conditions
We have a wire of length \( L \), cross-sectional area \( A \), and resistance \( R \). The resistance \( R \) of the wire is given by the formula:
\[ R = \rho \frac{L}{A} \]
where \( \rho \) is the resistivity of the material.

### Step 2: Determine the circumference of the circle
When the wire is bent into a complete circle, the length of the wire becomes the circumference of the circle:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

### Step 3: Relate the length of the wire to the radius
Since the wire was originally of length \( L \), we have:
\[ L = 2\pi r \]
From this, we can express the radius \( r \):
\[ r = \frac{L}{2\pi} \]

### Step 4: Determine the resistance of each semicircle
When the wire is bent into a circle, it can be divided into two semicircular sections. Each semicircular section has a length of:
\[ \text{Length of each semicircle} = \frac{L}{2} \]
The resistance of each semicircular section can be calculated using the resistance formula:
\[ R_{\text{semicircle}} = \rho \frac{\frac{L}{2}}{A} = \frac{R}{2} \]
Thus, each semicircle has a resistance of \( \frac{R}{2} \).

### Step 5: Calculate the equivalent resistance between two diametrically opposite points
The two semicircles are in parallel when considering the resistance between points A and B (the diametrically opposite points). The equivalent resistance \( R_{AB} \) can be calculated using the formula for resistors in parallel:
\[ \frac{1}{R_{AB}} = \frac{1}{R_{\text{semicircle}}} + \frac{1}{R_{\text{semicircle}}} \]
Substituting \( R_{\text{semicircle}} = \frac{R}{2} \):
\[ \frac{1}{R_{AB}} = \frac{1}{\frac{R}{2}} + \frac{1}{\frac{R}{2}} = \frac{2}{R} + \frac{2}{R} = \frac{4}{R} \]
Thus,
\[ R_{AB} = \frac{R}{4} \]

### Final Answer
The resistance between two diametrically opposite points is:
\[ R_{AB} = \frac{R}{4} \]