An electric dipole is kept on the axis of a uniformly charged ring at distance d from the centre of the ring. The direction of the dipole moment is along the axis. The dipole moment is `p`, charge of the ring is `Q` & radius of the ring is `R`. The force on the dipole is

Options

A. `(pQ)/(3piepsilon_(0)sqrt(3)R^(2))`
B. `(4pQ)/(3piepsilon_(0)sqrt(3)R^(2))`
C. `(pQ)/(3piepsilon_(0)R^(2))`
D. zero

Explanation

To solve the problem of finding the force on an electric dipole placed on the axis of a uniformly charged ring, we can follow these steps:

### Step 1: Understand the Setup
We have an electric dipole consisting of two charges, +q and -q, separated by a small distance (2L), placed at a distance \(d\) from the center of a uniformly charged ring of charge \(Q\) and radius \(R\). The dipole moment \(p\) is directed along the axis of the ring.

### Step 2: Calculate the Electric Field of the Ring
To find the force on the dipole, we first need to calculate the electric field \(E\) at the position of the dipole due to the charged ring. The electric field along the axis of a uniformly charged ring is given by the formula:

\[ E = \frac{kQd}{(R^2 + d^2)^{3/2}} \]

where \(k\) is Coulomb's constant, \(Q\) is the total charge of the ring, \(d\) is the distance from the center of the ring to the dipole, and \(R\) is the radius of the ring.

### Step 3: Determine the Forces on the Charges of the Dipole
The dipole consists of two charges:
- The positive charge \(+q\) experiences a force \(F_+\) in the direction of the electric field:
\[ F_+ = +qE \]
- The negative charge \(-q\) experiences a force \(F_-\) in the opposite direction:
\[ F_- = -qE \]

### Step 4: Calculate the Net Force on the Dipole
The net force \(F\) acting on the dipole can be calculated as the difference between the forces acting on the two charges:
\[ F = F_+ + F_- = qE - qE = 0 \]

### Conclusion
Since the forces on the two charges of the dipole are equal in magnitude and opposite in direction, the net force acting on the dipole is zero.

### Final Answer
The force on the dipole is **zero**.

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