Two point dipoles of dipole moment `vec(p)_(1) and vec(p)_(2)` are at a distance x from each other and `vec(p)_(1)||vec(p)_(2)` . The force between the dipoles is :

Options

A. `(1)/(4 pi epsilon_(0))(4 p_(1)p_(2))/(x^(4))`
B. `(1)/(4 pi epsilon_(0))(3 p_(1)p_(2))/(x^(3))`
C. `(1)/(4 pi epsilon_(0))(6 p_(1)p_(2))/(x^(4))`
D. `(1)/(4 pi epsilon_(0))(8 p_(1)p_(2))/(x^(4))`

Answer

To solve the problem of finding the force between two point dipoles with dipole moments \(\vec{p}_1\) and \(\vec{p}_2\) that are at a distance \(x\) from each other and are parallel, we can follow these steps:

### Step 1: Understand the Electric Field due to a Dipole
The electric field \(E\) at a point due to a dipole moment \(\vec{p}\) at a distance \(x\) along the axis of the dipole is given by the formula:
\[ E = \frac{1}{4 \pi \epsilon_0} \cdot \frac{2p}{x^3} \]
where \(p\) is the magnitude of the dipole moment.

### Step 2: Calculate the Electric Field due to Dipole 1
For dipole 1 with dipole moment \(\vec{p}_1\), the electric field \(E_1\) at the location of dipole 2 (which is at distance \(x\) from dipole 1) is:
\[ E_1 = \frac{1}{4 \pi \epsilon_0} \cdot \frac{2p_1}{x^3} \]

### Step 3: Calculate the Force on Dipole 2
The force \(F\) on a dipole in an electric field is given by:
\[ F = \vec{p} \cdot \frac{d\vec{E}}{dx} \]
For dipole 2 with dipole moment \(\vec{p}_2\), the force can be expressed as:
\[ F = p_2 \cdot \frac{dE_1}{dx} \]

### Step 4: Differentiate the Electric Field
Now we need to differentiate \(E_1\) with respect to \(x\):
\[ E_1 = \frac{1}{4 \pi \epsilon_0} \cdot \frac{2p_1}{x^3} \]
Taking the derivative:
\[ \frac{dE_1}{dx} = \frac{d}{dx} \left(\frac{1}{4 \pi \epsilon_0} \cdot \frac{2p_1}{x^3}\right) \]
Using the power rule for differentiation:
\[ \frac{dE_1}{dx} = \frac{1}{4 \pi \epsilon_0} \cdot 2p_1 \cdot \left(-3x^{-4}\right) = -\frac{6p_1}{4 \pi \epsilon_0 x^4} \]

### Step 5: Substitute Back to Find the Force
Now substitute \(\frac{dE_1}{dx}\) back into the force equation:
\[ F = p_2 \cdot \left(-\frac{6p_1}{4 \pi \epsilon_0 x^4}\right) \]
This simplifies to:
\[ F = -\frac{6p_1 p_2}{4 \pi \epsilon_0 x^4} \]
The negative sign indicates that the force is attractive if the dipoles are aligned.

### Conclusion
Thus, the magnitude of the force between the two dipoles is:
\[ F = \frac{6p_1 p_2}{4 \pi \epsilon_0 x^4} \]

### Final Answer
The correct option is:
\[ \frac{1}{4 \pi \epsilon_0} \cdot 6 \frac{p_1 p_2}{x^4} \]