What is the magnitude of electric intensity due to a dipole of moment `2xx10^(-8)C-m` at a point distant `1 m` from the centre of dipole, when line joining the point to the center of dipole makes an angle of `60^(@)` with diople exis ?

Options

A. `191*1 N//C`
B. `238*1 N//C`
C. `400 N//C`
D. `840 N//C`

Explanation

To find the magnitude of electric intensity (electric field) due to a dipole at a given point, we can use the formula for the electric field due to a dipole:

\[ E = \frac{1}{4 \pi \epsilon_0} \cdot \frac{P}{R^3} \cdot \sqrt{3 \cos^2 \theta + 1} \]

Where:
- \(E\) is the electric field intensity,
- \(P\) is the dipole moment,
- \(R\) is the distance from the dipole,
- \(\theta\) is the angle between the dipole axis and the line joining the dipole to the point of interest,
- \(\epsilon_0\) is the permittivity of free space, approximately \(8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2\).

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Dipole moment, \(P = 2 \times 10^{-8} \, \text{C m}\)
- Distance from the dipole, \(R = 1 \, \text{m}\)
- Angle, \(\theta = 60^\circ\)

2. **Convert the Angle to Cosine:**
- Calculate \(\cos 60^\circ\):
\[ \cos 60^\circ = \frac{1}{2} \]

3. **Substitute Values into the Electric Field Formula:**
- Plugging in the values into the electric field formula:
\[ E = \frac{1}{4 \pi \epsilon_0} \cdot \frac{P}{R^3} \cdot \sqrt{3 \cos^2 60^\circ + 1} \]

4. **Calculate \(\cos^2 60^\circ\):**
- Calculate \(\cos^2 60^\circ\):
\[ \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

5. **Calculate the Expression Inside the Square Root:**
- Substitute \(\cos^2 60^\circ\) into the expression:
\[ 3 \cos^2 60^\circ + 1 = 3 \cdot \frac{1}{4} + 1 = \frac{3}{4} + 1 = \frac{3}{4} + \frac{4}{4} = \frac{7}{4} \]

6. **Calculate the Electric Field:**
- Substitute \(P\), \(R\), and the calculated values into the formula:
\[ E = \frac{1}{4 \pi (8.85 \times 10^{-12})} \cdot \frac{2 \times 10^{-8}}{1^3} \cdot \sqrt{\frac{7}{4}} \]
- Calculate \(E\):
\[ E = \frac{1}{4 \pi (8.85 \times 10^{-12})} \cdot 2 \times 10^{-8} \cdot \sqrt{\frac{7}{4}} \]
- Calculate \(\sqrt{\frac{7}{4}} = \frac{\sqrt{7}}{2}\):
\[ E = \frac{1}{4 \pi (8.85 \times 10^{-12})} \cdot 2 \times 10^{-8} \cdot \frac{\sqrt{7}}{2} \]
- Simplifying gives:
\[ E = \frac{1}{4 \pi (8.85 \times 10^{-12})} \cdot 10^{-8} \cdot \sqrt{7} \]
- Using \(4 \pi \epsilon_0 \approx 9 \times 10^9 \, \text{N m}^2/\text{C}^2\):
\[ E \approx \frac{9 \times 10^9}{10^{-8}} \cdot \sqrt{7} \approx 38.1 \, \text{N/C} \]

### Final Answer:
The magnitude of electric intensity due to the dipole at the given point is approximately \(38.1 \, \text{N/C}\).