A dielectric slab of thickness `d` is inserted in a parallel plate capacitor whose negative plate is at `x=0` and positive plate is at `x = 3d`. The slab is equidistant from the plates. The capacitor is given some charge. As one goes from `0` to `3d(1998)`.

Options

A. the magnitude of the electric field remains the same
B. the direction of the electric remains the same
C. the electric potential increases continuously
D. the electric potential increases at first, then decreases and again increases

Answer

To solve the problem step by step, let's analyze the situation involving the parallel plate capacitor with a dielectric slab inserted between the plates.

### Step 1: Understand the Configuration
We have a parallel plate capacitor with:
- Negative plate at \( x = 0 \)
- Positive plate at \( x = 3d \)
- A dielectric slab of thickness \( d \) inserted equidistantly between the plates.

### Step 2: Identify the Electric Field
The electric field \( E \) in a parallel plate capacitor is given by the formula:
\[ E = \frac{V}{d} \]
where \( V \) is the potential difference between the plates and \( d \) is the separation between them. The direction of the electric field is from the positive plate to the negative plate.

### Step 3: Determine the Effect of the Dielectric
When a dielectric slab is inserted, it affects the electric field. However, since the slab is equidistant from both plates, the electric field will still point from the positive plate to the negative plate, but its magnitude will change depending on the dielectric constant \( \kappa \) of the slab.

### Step 4: Analyze the Electric Potential
The electric potential \( V \) at a distance \( x \) from the negative plate can be expressed as:
\[ V(x) = -\int E \, dx \]
Since the electric field is directed towards the negative plate, the potential increases as we move from the negative plate to the positive plate.

### Step 5: Evaluate the Options
1. **Magnitude of the Electric Field**: The magnitude of the electric field does not remain the same due to the presence of the dielectric. Thus, this option is incorrect.
2. **Direction of the Electric Field**: The direction of the electric field remains the same (from positive to negative). This option is correct.
3. **Electric Potential**: The electric potential increases continuously as we move from the negative plate to the positive plate. This option is correct.
4. **Electric Potential Behavior**: The electric potential does not increase and then decrease; it continuously increases. Thus, this option is incorrect.

### Final Conclusion
The correct options are:
- **B**: The direction of the electric field remains the same.
- **C**: The electric potential increases continuously.