The electric potential in a region is given by the relation `V(x) = 4+5x^2`. If a dipole is placed at position (-1,0) with dipole moment `vecP` pointing along positive y-direction, then

Options

A. net force on the dipole is zero.
B. net torque on the dipole is zero.
C. net torque on the dipole is not zero and it is in clockwise direction.
D. Net torque on the dipole is not zero and it is in anticlockwise direction.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Find the Electric Field
The electric potential \( V(x) \) is given by:
\[ V(x) = 4 + 5x^2 \]

The electric field \( E \) is related to the electric potential by the equation:
\[ E = -\frac{dV}{dx} \]

Calculating the derivative:
\[ \frac{dV}{dx} = \frac{d}{dx}(4 + 5x^2) = 10x \]

Thus, the electric field is:
\[ E = -10x \]

### Step 2: Evaluate the Electric Field at the Dipole's Position
The dipole is placed at position \( (-1, 0) \):
\[ E(-1) = -10(-1) = 10 \, \text{N/C} \]
The electric field at the position of the dipole is \( 10 \, \text{N/C} \) in the positive x-direction.

### Step 3: Determine the Torque on the Dipole
The torque \( \tau \) on a dipole in an electric field is given by:
\[ \tau = \vec{P} \times \vec{E} \]

Given that the dipole moment \( \vec{P} \) is directed along the positive y-direction and \( \vec{E} \) is directed along the positive x-direction, the angle \( \theta \) between \( \vec{P} \) and \( \vec{E} \) is \( 90^\circ \).

The magnitude of the torque is:
\[ |\tau| = P \cdot E \cdot \sin(90^\circ) = P \cdot E \]

Since \( \sin(90^\circ) = 1 \), the torque is maximized.

### Step 4: Determine the Potential Energy of the Dipole
The potential energy \( U \) of a dipole in an electric field is given by:
\[ U = -\vec{P} \cdot \vec{E} \]

Since \( \vec{P} \) and \( \vec{E} \) are perpendicular, the potential energy becomes:
\[ U = -P \cdot E \cdot \cos(90^\circ) = 0 \]

### Step 5: Analyze the Options
- **Option A**: The net force on the dipole is zero. This is true because the forces on the positive and negative charges of the dipole are equal and opposite.
- **Option B**: The net torque on the dipole is zero. This is false because there is a non-zero torque acting on the dipole.
- **Option C**: The net torque on the dipole is not zero and is in the clockwise direction. This is true, as the torque is maximized and acts to rotate the dipole.
- **Option D**: The net torque acting is in the counterclockwise direction. This is false.

### Conclusion
The correct options are A and C.

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