An electric dipole is placed at the origin along `x`-axis.The electric field at any point , whose position vector , makes an angle `theta` with the `x`-axis ,makes an angle,

Options

A. `alpha`
B. `theta`
C. `theta+alpha`
D. `theta+2alpha`


with the x-axis where `tan alpha` = `tan theta/2`

Answer

To solve the problem of finding the angle made by the electric field at point P with respect to the x-axis when an electric dipole is placed at the origin along the x-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have an electric dipole placed at the origin along the x-axis.
- The position vector of point P makes an angle θ with the x-axis.

2. **Drawing the Diagram**:
- Draw the x-axis and mark the origin where the dipole is located.
- Draw the position vector \( \vec{r} \) from the origin to point P, making an angle θ with the x-axis.

3. **Electric Field due to the Dipole**:
- The electric field \( \vec{E} \) at point P due to the dipole can be expressed using the formula:
\[ E = \frac{p}{4 \pi \epsilon_0} \cdot \frac{1}{r^3} \sqrt{3 \cos^2 \theta + 1} \]
where \( p \) is the dipole moment, \( r \) is the distance from the dipole to point P, and \( \epsilon_0 \) is the permittivity of free space.

4. **Angle Between Electric Field and Position Vector**:
- The electric field \( \vec{E} \) at point P makes an angle α with the position vector \( \vec{r} \).
- By definition, the angle between the electric field and the x-axis can be expressed as:
\[ \text{Angle with x-axis} = \theta + \alpha \]

5. **Finding the Relationship Between Angles**:
- We are given that \( \tan \alpha = \frac{\tan \theta}{2} \).
- This relationship will help us determine the angle α in terms of θ.

6. **Conclusion**:
- Therefore, the angle made by the electric field with respect to the x-axis is:
\[ \text{Angle} = \theta + \alpha \]
- Hence, the correct answer is \( \theta + \alpha \).

### Final Answer:
The electric field at point P makes an angle \( \theta + \alpha \) with the x-axis.

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