Two thin wire rings each having radius R are placed at distance d apart with their axes coinciding. The charges on the two are `+Q` and `-Q`. The potential difference between the centre so the two rings is

Options

A. zero
B. `Q/(4piepsilon_0) [1/R-1/(sqrt(R^2+d^2))]`
C. `Q/4piepsilon_0d^2`
D. `Q/(2piepsilon_0)[1/R-1/(sqrt(R^2+d^2))]`

Explanation

To find the potential difference between the centers of two thin wire rings with charges +Q and -Q, we can follow these steps:

### Step 1: Understand the Configuration
We have two thin wire rings, each with radius \( R \), separated by a distance \( d \). One ring has a charge of \( +Q \) and the other has a charge of \( -Q \). We need to find the potential difference between the centers of these two rings.

### Step 2: Calculate the Potential at the Center of the First Ring
The potential \( V_A \) at the center of the first ring (with charge \( +Q \)) is given by the formula:
\[ V_A = \frac{kQ}{R} \]
where \( k \) is the Coulomb's constant.

### Step 3: Calculate the Distance from the Center of the First Ring to the Second Ring
To find the potential due to the second ring at the center of the first ring, we need to calculate the distance from the center of the first ring to the center of the second ring. This distance can be found using the Pythagorean theorem:
\[ \text{Distance} = \sqrt{R^2 + d^2} \]

### Step 4: Calculate the Potential at the Center of the Second Ring
The potential \( V_B \) at the center of the second ring (with charge \( -Q \)) is given by:
\[ V_B = \frac{-kQ}{\sqrt{R^2 + d^2}} \]

### Step 5: Calculate the Potential Difference
The potential difference \( V_{AB} \) between the centers of the two rings is given by:
\[ V_{AB} = V_A - V_B \]
Substituting the values we found:
\[ V_{AB} = \frac{kQ}{R} - \left(-\frac{kQ}{\sqrt{R^2 + d^2}}\right) \]
This simplifies to:
\[ V_{AB} = \frac{kQ}{R} + \frac{kQ}{\sqrt{R^2 + d^2}} \]

### Step 6: Factor Out Common Terms
Factoring out \( kQ \) gives us:
\[ V_{AB} = kQ \left( \frac{1}{R} + \frac{1}{\sqrt{R^2 + d^2}} \right) \]

### Final Result
Thus, the potential difference between the centers of the two rings is:
\[ V_{AB} = kQ \left( \frac{1}{R} - \frac{1}{\sqrt{R^2 + d^2}} \right) \]