An electric dipole is placed at origin in the xy plane with its orientation along the positive x-axis. The direction is electric field .

Options

A. at `(-x_0,0,0)` is along the positive x-axis
B. at (0,`y_0`,0) is along the negative x-axis.
C. at `(0,0,z_0)` is along the negative x-axis
D. all the above

Explanation

To determine the direction of the electric field created by an electric dipole placed at the origin in the xy-plane with its orientation along the positive x-axis, we can follow these steps:

### Step 1: Understand the Electric Dipole
An electric dipole consists of two equal and opposite charges separated by a distance \(d\). The dipole moment \( \vec{p} \) is given by:
\[ \vec{p} = q \cdot \vec{d} \]
where \(q\) is the magnitude of one of the charges and \( \vec{d} \) is the vector pointing from the negative charge to the positive charge.

### Step 2: Analyze the Orientation
Given that the dipole is oriented along the positive x-axis, the dipole moment \( \vec{p} \) points in the positive x-direction.

### Step 3: Electric Field Due to a Dipole
The electric field \( \vec{E} \) due to a dipole at a point in space can be calculated using the formula:
\[ \vec{E} = \frac{1}{4\pi \epsilon_0} \cdot \frac{2\vec{p} \cdot \hat{r}}{r^3} - \frac{p}{4\pi \epsilon_0 r^3} \hat{r} \]
where \( \hat{r} \) is the unit vector in the direction from the dipole to the point where the field is being calculated, and \( r \) is the distance from the dipole to that point.

### Step 4: Evaluate at Given Points
Let's evaluate the electric field direction at the specified points:

- **Point A: (-x0, 0, 0)**
- The position vector points towards the negative x-axis. The electric field direction will be towards the positive x-axis, as the field lines point away from the positive charge and towards the negative charge.

- **Point B: (0, y0, 0)**
- The position vector points along the positive y-axis. The electric field direction at this point will be towards the negative x-axis, as the dipole's field lines will pull towards the negative charge.

- **Point C: (0, -y0, 0)**
- Similar to point B, the position vector points along the negative y-axis. The electric field direction will also be towards the negative x-axis.

### Step 5: Conclusion
From the analysis:
- At point A, the electric field is towards the positive x-axis.
- At points B and C, the electric field is towards the negative x-axis.

Thus, the correct answer is that all options (A, B, and C) are correct, leading to option D: all of the above.