The force between two short electric dipoles separated by a distance r is direcly proportional to :

Options

A. `(1)/( R )`
B. `(1)/(R^(2))`
C. `(1)/(R^(3))`
D. `(1)/( R^(4))`

Answer

To solve the question regarding the force between two short electric dipoles separated by a distance \( r \), we can follow these steps:

### Step 1: Understand the Electric Dipole
An electric dipole consists of two equal and opposite charges separated by a small distance. The dipole moment \( \mathbf{P} \) is defined as:
\[ \mathbf{P} = q \cdot \mathbf{d} \]
where \( q \) is the magnitude of one of the charges and \( \mathbf{d} \) is the vector pointing from the negative charge to the positive charge.

### Step 2: Determine the Electric Field Due to a Dipole
The electric field \( \mathbf{E} \) at a point along the axial line of a dipole at a distance \( r \) from its center is given by:
\[ \mathbf{E} = \frac{1}{4\pi \epsilon_0} \cdot \frac{2\mathbf{P}}{r^3} \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 3: Calculate the Force on the Second Dipole
When a second dipole \( \mathbf{P_2} \) is placed in the electric field \( \mathbf{E} \) created by the first dipole \( \mathbf{P_1} \), the force \( \mathbf{F} \) acting on the second dipole can be expressed as:
\[ \mathbf{F} = \mathbf{P_2} \cdot \mathbf{E} \]
Substituting the expression for \( \mathbf{E} \):
\[ \mathbf{F} = \mathbf{P_2} \cdot \left(\frac{1}{4\pi \epsilon_0} \cdot \frac{2\mathbf{P_1}}{r^3}\right) \]

### Step 4: Express the Force in Terms of \( r \)
The magnitude of the force can be simplified to:
\[ \mathbf{F} = \frac{2}{4\pi \epsilon_0} \cdot \frac{\mathbf{P_1} \cdot \mathbf{P_2}}{r^3} \]
This shows that the force between the two dipoles is inversely proportional to the cube of the distance \( r \).

### Step 5: Conclusion
Thus, the force between two short electric dipoles separated by a distance \( r \) is directly proportional to the product of their dipole moments and inversely proportional to the fourth power of the distance \( r \):
\[ \mathbf{F} \propto \frac{\mathbf{P_1} \cdot \mathbf{P_2}}{r^4} \]

### Final Answer
The force between two short electric dipoles separated by a distance \( r \) is directly proportional to \( \frac{1}{r^4} \).

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