Electric dipole of moment `vec(p)=p hat(i)` is kept at a point (x, y) in an electric field `vec(E )=4x y^(2)hat(i)+4x^(2)y hat(j)`. Find the magnitude of force acting on the dipole.

Options

A. `4py sqrt(y^(2)+2x^(2))`
B. `2py sqrt(y^(2)+4x^(2))`
C. `4py sqrt(y^(2)+4x^(2))`
D. `2py sqrt(y^(2)+2x^(2))`

Answer

To find the magnitude of the force acting on the electric dipole in the given electric field, we can follow these steps:

### Step 1: Understand the Given Information
We have an electric dipole moment \(\vec{p} = p \hat{i}\) and an electric field \(\vec{E} = 4xy^2 \hat{i} + 4x^2y \hat{j}\).

### Step 2: Determine the Force on the Dipole
The force \(\vec{F}\) acting on an electric dipole in a non-uniform electric field is given by the formula:
\[ \vec{F} = \vec{p} \cdot \nabla \vec{E} \]
where \(\nabla \vec{E}\) is the gradient of the electric field.

### Step 3: Compute the Gradient of the Electric Field
The electric field components are:
- \(E_x = 4xy^2\)
- \(E_y = 4x^2y\)

We need to find the derivatives of these components with respect to \(x\) and \(y\).

1. **Partial derivative of \(E_x\) with respect to \(x\)**:
\[ \frac{\partial E_x}{\partial x} = \frac{\partial}{\partial x}(4xy^2) = 4y^2 \]

2. **Partial derivative of \(E_x\) with respect to \(y\)**:
\[ \frac{\partial E_x}{\partial y} = \frac{\partial}{\partial y}(4xy^2) = 8xy \]

3. **Partial derivative of \(E_y\) with respect to \(x\)**:
\[ \frac{\partial E_y}{\partial x} = \frac{\partial}{\partial x}(4x^2y) = 8xy \]

4. **Partial derivative of \(E_y\) with respect to \(y\)**:
\[ \frac{\partial E_y}{\partial y} = \frac{\partial}{\partial y}(4x^2y) = 4x^2 \]

### Step 4: Formulate the Resultant Force
Now we can express the force \(\vec{F}\) using the dipole moment \(\vec{p} = p \hat{i}\):
\[ \vec{F} = p \left( \frac{\partial E_x}{\partial x} \hat{i} + \frac{\partial E_y}{\partial y} \hat{j} \right) \]
Substituting the partial derivatives:
\[ \vec{F} = p \left( 4y^2 \hat{i} + 4x^2 \hat{j} \right) \]

### Step 5: Calculate the Magnitude of the Force
The magnitude of the force \(|\vec{F}|\) can be calculated as:
\[ |\vec{F}| = \sqrt{(p \cdot 4y^2)^2 + (p \cdot 4x^2)^2} \]
\[ |\vec{F}| = p \sqrt{(4y^2)^2 + (4x^2)^2} \]
\[ |\vec{F}| = p \sqrt{16y^4 + 16x^4} \]
\[ |\vec{F}| = 4p \sqrt{y^4 + x^4} \]

### Final Answer
The magnitude of the force acting on the dipole is:
\[ |\vec{F}| = 4p \sqrt{y^4 + x^4} \]