An electric field of `20 N//C` exists along the x-axis in space. Calculate the potential difference `V_B - V_A` where the points A and B are given by
a. `A=(0,0), B=(4m, 2m)`
b. `A=(4m,2m),B=(6m,5m)`

Explanation

To solve the problem of calculating the potential difference \( V_B - V_A \) for the given electric field and points, we will follow these steps:

### Part (a): Points A = (0, 0) and B = (4 m, 2 m)

1. **Identify the Electric Field**:
The electric field \( \mathbf{E} \) is given as \( 20 \, \text{N/C} \) along the x-axis. This can be represented as:
\[ \mathbf{E} = 20 \, \hat{i} \, \text{N/C} \]

2. **Set Up the Integral for Potential Difference**:
The potential difference \( V_B - V_A \) can be calculated using the formula:
\[ V_B - V_A = -\int_{A}^{B} \mathbf{E} \cdot d\mathbf{r} \]
Here, \( d\mathbf{r} \) is the differential displacement vector.

3. **Parameterize the Path**:
Since the electric field is only in the x-direction, we can simplify the integral. The displacement vector \( d\mathbf{r} \) can be expressed as \( dx \, \hat{i} + dy \, \hat{j} \). However, since the electric field does not have a y-component, we will only consider the x-component for the calculation.

4. **Determine the Limits of Integration**:
The limits for \( x \) will be from \( 0 \) to \( 4 \) (the x-coordinate of point B). The y-coordinate does not affect the potential difference in this case.

5. **Evaluate the Integral**:
The integral simplifies to:
\[ V_B - V_A = -\int_{0}^{4} 20 \, dx = -20 \left[ x \right]_{0}^{4} = -20(4 - 0) = -80 \, \text{V} \]

### Part (b): Points A = (4 m, 2 m) and B = (6 m, 5 m)

1. **Identify the Electric Field**:
The electric field remains the same:
\[ \mathbf{E} = 20 \, \hat{i} \, \text{N/C} \]

2. **Set Up the Integral for Potential Difference**:
Again, we use the formula:
\[ V_B - V_A = -\int_{A}^{B} \mathbf{E} \cdot d\mathbf{r} \]

3. **Parameterize the Path**:
For this part, we will consider the x-component of the path from \( x = 4 \) to \( x = 6 \). The y-component does not contribute to the potential difference.

4. **Determine the Limits of Integration**:
The limits for \( x \) will be from \( 4 \) to \( 6 \).

5. **Evaluate the Integral**:
The integral becomes:
\[ V_B - V_A = -\int_{4}^{6} 20 \, dx = -20 \left[ x \right]_{4}^{6} = -20(6 - 4) = -40 \, \text{V} \]

### Final Answers:
- For part (a): \( V_B - V_A = -80 \, \text{V} \)
- For part (b): \( V_B - V_A = -40 \, \text{V} \)