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.

Answer

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

### Step 1: Understand the Given Information
We have an electric dipole with a dipole moment:
\[ \vec{P} = P \hat{i} \]
and an electric field given by:
\[ \vec{E} = 4xy^2 \hat{i} + 4x^2y \hat{j} \]

### Step 2: Calculate the Electric Field Gradient
The force acting on the dipole in a non-uniform electric field is given by:
\[ \vec{F} = \vec{P} \cdot \nabla \vec{E} \]
where \(\nabla \vec{E}\) is the gradient of the electric field. We need to find the partial derivative of \(\vec{E}\) with respect to \(x\), treating \(y\) as a constant.

### Step 3: Differentiate the Electric Field
We differentiate \(\vec{E}\) with respect to \(x\):
\[ \frac{\partial \vec{E}}{\partial x} = \frac{\partial}{\partial x}(4xy^2 \hat{i} + 4x^2y \hat{j}) \]
Calculating the derivatives:
- For the \(\hat{i}\) component:
\[ \frac{\partial}{\partial x}(4xy^2) = 4y^2 \]
- For the \(\hat{j}\) component:
\[ \frac{\partial}{\partial x}(4x^2y) = 8xy \]

Thus, we have:
\[ \frac{\partial \vec{E}}{\partial x} = (4y^2 \hat{i} + 8xy \hat{j}) \]

### Step 4: Calculate the Force
Now, substituting \(\vec{P}\) and \(\frac{\partial \vec{E}}{\partial x}\) into the force equation:
\[ \vec{F} = \vec{P} \cdot \frac{\partial \vec{E}}{\partial x} = P \hat{i} \cdot (4y^2 \hat{i} + 8xy \hat{j}) \]
This simplifies to:
\[ \vec{F} = P(4y^2 \hat{i}) + P(0 \hat{j}) = 4Py^2 \hat{i} + 8Pxy \hat{j} \]

### Step 5: Find the Magnitude of the Force
To find the magnitude of the force vector \(\vec{F}\):
\[ |\vec{F}| = \sqrt{(4Py^2)^2 + (8Pxy)^2} \]
Calculating this:
\[ |\vec{F}| = \sqrt{16P^2y^4 + 64P^2x^2y^2} \]
Factoring out \(16P^2\):
\[ |\vec{F}| = 4P \sqrt{y^4 + 4x^2y^2} \]
We can further simplify:
\[ |\vec{F}| = 4Py \sqrt{y^2 + 4x^2} \]

### Final Answer
Thus, the magnitude of the force acting on the dipole is:
\[ |\vec{F}| = 4Py \sqrt{y^2 + 4x^2} \]

---