In a certain region of space, electric field is along the z-direction throughout. The magnitude of electric field is , however, not constant but increases uniformly along the positive z-direction. At the rate of `10^(5)NC^(-1)m^(-1)` . What are the force and torque experienced by system having a total dipole moment equal to `10^(-7)`Cm in the negative z-direction?

Options

A. `-10^(-2)N`
B. `10^(-2)N`
C. `10^(-4)N`
D. `-10^(-4)N`

Explanation

To solve the problem, we need to calculate the force and torque experienced by a dipole in a non-uniform electric field. The steps are as follows:

### Step 1: Identify the given values
- The rate of increase of the electric field \( \frac{dE}{dz} = 10^5 \, \text{N/C/m} \)
- The dipole moment \( P = -10^{-7} \, \text{C m} \) (in the negative z-direction)

### Step 2: Calculate the force on the dipole
The force \( F \) on a dipole in a non-uniform electric field can be calculated using the formula:
\[ F = P \frac{dE}{dz} \]
Substituting the values:
\[ F = (-10^{-7} \, \text{C m}) \times (10^5 \, \text{N/C/m}) \]
\[ F = -10^{-2} \, \text{N} \]
This indicates that the force is directed in the negative z-direction.

### Step 3: Calculate the torque on the dipole
The torque \( \tau \) on a dipole in an electric field is given by:
\[ \tau = P E \sin(\theta) \]
Where \( \theta \) is the angle between the dipole moment and the electric field. Since the dipole moment is in the negative z-direction and the electric field is in the positive z-direction, the angle \( \theta = 180^\circ \).

Substituting the values:
\[ \tau = (-10^{-7} \, \text{C m}) \times E \times \sin(180^\circ) \]
Since \( \sin(180^\circ) = 0 \):
\[ \tau = 0 \]

### Final Results
- The force experienced by the dipole is \( -10^{-2} \, \text{N} \) (in the negative z-direction).
- The torque experienced by the dipole is \( 0 \).

### Summary
- **Force**: \( -10^{-2} \, \text{N} \)
- **Torque**: \( 0 \)