The electric field of an electric dipole at a point on its axis , at a distance d from the center of the dipole, varies as

Options

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

Answer

To solve the problem of how the electric field of an electric dipole varies at a point on its axis at a distance \( d \) from the center of the dipole, we can follow these steps:

### Step 1: Understand the Concept of an Electric Dipole
An electric dipole consists of two equal and opposite charges separated by a distance \( 2a \). The dipole moment \( P \) is defined as:
\[ P = q \cdot 2a \]
where \( q \) is the magnitude of one of the charges.

### Step 2: Electric Field on the Axis of the Dipole
The electric field \( E \) at a point on the axis of the dipole, at a distance \( d \) from the center of the dipole, can be derived from the formula:
\[ E = \frac{1}{4\pi \epsilon_0} \cdot \frac{2P}{d^3} \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 3: Analyze the Formula
From the formula, we can see that:
\[ E \propto \frac{P}{d^3} \]
This indicates that the electric field \( E \) is directly proportional to the dipole moment \( P \) and inversely proportional to the cube of the distance \( d \).

### Step 4: Conclusion
Thus, the electric field of an electric dipole at a point on its axis at a distance \( d \) from the center of the dipole varies as:
\[ E \propto \frac{1}{d^3} \]

### Final Answer
The electric field of an electric dipole at a point on its axis at a distance \( d \) from the center of the dipole varies as \( \frac{1}{d^3} \).

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