The electric field due to a dipole at a distance on its axis is

Options

A. Directly proportional to `r^(3)`
B. Inversely proportional to `r^(3)`
C. Directly proportional to `r^(2)`
D. Inversely proportional to `r^(2)`

Answer

To find the electric field due to a dipole at a distance \( r \) on its axis, we can follow these steps:

### Step 1: Understand the Dipole Configuration
A dipole consists of two equal and opposite charges, \( +q \) and \( -q \), separated by a distance \( 2L \). The dipole moment \( p \) is defined as:
\[ p = q \cdot 2L \]
where \( p \) is a vector pointing from the negative charge to the positive charge.

### Step 2: Identify the Point of Interest
We are interested in the electric field at a point along the axis of the dipole, at a distance \( r \) from the center of the dipole.

### Step 3: Use the Formula for Electric Field on the Axis of a Dipole
The electric field \( E \) at a distance \( r \) on the axis of a dipole is given by the formula:
\[ E = \frac{2p}{4 \pi \epsilon_0 r^3} \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 4: Analyze the Dependence of the Electric Field
From the formula, we can see that:
- The electric field \( E \) is directly proportional to the dipole moment \( p \).
- The electric field \( E \) is inversely proportional to the cube of the distance \( r \) (i.e., \( E \propto \frac{1}{r^3} \)).

### Step 5: Conclusion
Thus, the electric field due to a dipole at a distance \( r \) on its axis is:
\[ E = \frac{2p}{4 \pi \epsilon_0 r^3} \]