An electric dipole of moment `overset( r ) (p) = (hat(i) + 2hat ( j )) xx10^(-28) Cm` is at origin. The electric field at point ( 2,4) due to the dipole is parallelto

Options

A. `hati+2hatj`
B. `2hati+2hatj`
C. `hati-2hatj`
D. `2hati-hatj`

Explanation

To solve the problem, we need to determine the electric field at the point (2, 4) due to the given electric dipole moment. Let's break down the solution step by step.

### Step 1: Identify the Electric Dipole Moment
The electric dipole moment is given as:
\[ \vec{p} = (\hat{i} + 2\hat{j}) \times 10^{-28} \, \text{Cm} \]

### Step 2: Determine the Position Vector
The position vector \(\vec{r}\) of the point (2, 4) with respect to the origin (where the dipole is located) is:
\[ \vec{r} = 2\hat{i} + 4\hat{j} \]

### Step 3: Check for Parallelism
To find out if the electric field at point (2, 4) is parallel to the dipole moment, we need to check if \(\vec{p}\) and \(\vec{r}\) are parallel. Two vectors are parallel if one is a scalar multiple of the other.

### Step 4: Express the Vectors
We can express both vectors:
- \(\vec{p} = \hat{i} + 2\hat{j}\)
- \(\vec{r} = 2\hat{i} + 4\hat{j}\)

### Step 5: Check for Scalar Multiples
We can factor out 2 from \(\vec{r}\):
\[ \vec{r} = 2(\hat{i} + 2\hat{j}) \]
This indicates that \(\vec{r}\) is a scalar multiple of \(\vec{p}\):
\[ \vec{r} = 2 \cdot \frac{1}{2} \vec{p} \]
Thus, \(\vec{p}\) and \(\vec{r}\) are parallel.

### Step 6: Conclusion about the Electric Field Direction
Since \(\vec{p}\) and \(\vec{r}\) are parallel, the electric field \(\vec{E}\) at the point (2, 4) due to the dipole will also be parallel to \(\vec{p}\).

### Step 7: Electric Field Direction
The electric field direction at an axial point (along the line of the dipole) is in the same direction as the dipole moment. Therefore, the electric field at point (2, 4) is parallel to:
\[ \vec{p} = \hat{i} + 2\hat{j} \]

### Final Answer
The electric field at point (2, 4) due to the dipole is parallel to \(\hat{i} + 2\hat{j}\).

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