Two short dipoles, each of diple moment p , are placed at origin. The dipole moment of one dipole is along x-axis, while that of other is along y-axis . The electric field at a point (a,0) is given by

Options

A. `((1)/( 4pi epsilon_(0))) (2p)/(a^(3))`
B. `((1)/( 4pi epsilon_(0))) (p)/(a^(3))`
C. `((1)/( 4pi epsilon_(0))) (sqrt(5)p)/(a^(3))`
D. Zero

Answer

To find the electric field at the point (a, 0) due to two short dipoles placed at the origin, we can follow these steps:

### Step 1: Understand the dipoles' configuration
We have two dipoles at the origin:
- Dipole 1 has a dipole moment \( \vec{p_1} \) along the x-axis.
- Dipole 2 has a dipole moment \( \vec{p_2} \) along the y-axis.

### Step 2: Write the formula for the electric field due to a dipole
The electric field \( \vec{E} \) at a point along the axial line of a dipole is given by:
\[ \vec{E} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{2\vec{p}}{r^3} \]
where \( \vec{p} \) is the dipole moment and \( r \) is the distance from the dipole to the point of interest.

### Step 3: Calculate the electric field due to each dipole at point (a, 0)
1. **Electric field due to dipole 1 (along x-axis)**:
- The distance from the dipole to point (a, 0) is \( r = a \).
- The electric field \( \vec{E_1} \) due to dipole 1 is:
\[ \vec{E_1} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{2p}{a^3} \hat{i} \]

2. **Electric field due to dipole 2 (along y-axis)**:
- The distance from the dipole to point (a, 0) is still \( r = a \).
- The electric field \( \vec{E_2} \) due to dipole 2 is:
\[ \vec{E_2} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{p}{a^3} \hat{j} \]

### Step 4: Combine the electric fields
Since the two dipoles are perpendicular to each other, we can find the resultant electric field \( \vec{E} \) by using the Pythagorean theorem:
\[ \vec{E} = \sqrt{E_1^2 + E_2^2} \]
Substituting the expressions for \( E_1 \) and \( E_2 \):
\[ E = \sqrt{\left(\frac{1}{4 \pi \epsilon_0} \cdot \frac{2p}{a^3}\right)^2 + \left(\frac{1}{4 \pi \epsilon_0} \cdot \frac{p}{a^3}\right)^2} \]
\[ = \sqrt{\left(\frac{2p}{4 \pi \epsilon_0 a^3}\right)^2 + \left(\frac{p}{4 \pi \epsilon_0 a^3}\right)^2} \]
\[ = \sqrt{\frac{4p^2}{(4 \pi \epsilon_0)^2 a^6} + \frac{p^2}{(4 \pi \epsilon_0)^2 a^6}} \]
\[ = \sqrt{\frac{5p^2}{(4 \pi \epsilon_0)^2 a^6}} = \frac{\sqrt{5}p}{4 \pi \epsilon_0 a^3} \]

### Step 5: Final expression for the electric field
Thus, the electric field at the point (a, 0) due to the two dipoles is:
\[ \vec{E} = \frac{\sqrt{5}p}{4 \pi \epsilon_0 a^3} \]