The force on a charge situated on the axis of a small dipole is `F` .If the charge is shifted to double the distance the force acting will be `(f)/(x)` .What is `x` ?

Options

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

Answer

To solve the problem, we need to analyze the relationship between the force acting on a charge due to a dipole and the distance from the dipole. Here are the steps to arrive at the solution:

### Step 1: Understand the Initial Setup
- We have a small electric dipole and a charge \( Q \) located at a distance \( R \) from the center of the dipole along its axial line.
- The force \( F \) acting on the charge \( Q \) due to the dipole can be expressed as:
\[ F = Q \cdot E \]
where \( E \) is the electric field due to the dipole at distance \( R \).

### Step 2: Electric Field of a Dipole
- The electric field \( E \) on the axial line of a dipole is given by:
\[ E \propto \frac{1}{R^3} \]
- Therefore, the force \( F \) can be expressed as:
\[ F \propto \frac{1}{R^3} \]

### Step 3: Shift the Charge to Double the Distance
- Now, if the charge is moved to double the distance, the new distance \( R' \) becomes:
\[ R' = 2R \]

### Step 4: Calculate the New Force
- The new electric field \( E' \) at distance \( R' \) is:
\[ E' \propto \frac{1}{(2R)^3} = \frac{1}{8R^3} \]
- Thus, the new force \( F' \) acting on the charge \( Q \) at this new distance is:
\[ F' \propto \frac{1}{(2R)^3} = \frac{F}{8} \]

### Step 5: Relate the New Force to the Original Force
- From the relationship derived, we have:
\[ F' = \frac{F}{8} \]
- This indicates that the new force \( F' \) is \( \frac{F}{x} \) where \( x = 8 \).

### Conclusion
- Therefore, the value of \( x \) is:
\[ x = 8 \]