Potential difference between two points `(V_(A) - V_(B))` in an electric field `E = (2 hat(i) - 4 hat(j)) N//C`, where `A = (0, 0)` and `B = (3m, 4m)` is

Options

A. `10 V`
B. `-10 V`
C. `16 V`
D. `-16 V`

Answer

To find the potential difference between points A and B in the given electric field, we can follow these steps:

### Step 1: Understand the Electric Field and Coordinates
The electric field is given as:
\[ \mathbf{E} = 2 \hat{i} - 4 \hat{j} \, \text{N/C} \]
Point A has coordinates \( A(0, 0) \) and point B has coordinates \( B(3 \, \text{m}, 4 \, \text{m}) \).

### Step 2: Set Up the Potential Difference Formula
The potential difference \( V_A - V_B \) can be calculated using the formula:
\[ V_A - V_B = - \int_A^B \mathbf{E} \cdot d\mathbf{r} \]
where \( d\mathbf{r} \) is the differential displacement vector.

### Step 3: Define the Displacement Vector
The displacement vector \( d\mathbf{r} \) can be expressed in terms of its components:
\[ d\mathbf{r} = dx \hat{i} + dy \hat{j} \]

### Step 4: Calculate the Dot Product
Now, calculate the dot product \( \mathbf{E} \cdot d\mathbf{r} \):
\[ \mathbf{E} \cdot d\mathbf{r} = (2 \hat{i} - 4 \hat{j}) \cdot (dx \hat{i} + dy \hat{j}) = 2dx - 4dy \]

### Step 5: Set Up the Integral
Substituting the dot product into the potential difference equation gives:
\[ V_A - V_B = - \int_A^B (2dx - 4dy) \]

### Step 6: Determine the Path of Integration
To integrate from point A to point B, we can move in the x-direction first and then in the y-direction. The limits for \( x \) will be from 0 to 3, and for \( y \) from 0 to 4.

### Step 7: Integrate the Expression
The integral can be split into two parts:
\[ V_A - V_B = - \left( \int_0^3 2 \, dx + \int_0^4 (-4) \, dy \right) \]

Calculating the first integral:
\[ \int_0^3 2 \, dx = 2x \bigg|_0^3 = 2(3) - 2(0) = 6 \]

Calculating the second integral:
\[ \int_0^4 -4 \, dy = -4y \bigg|_0^4 = -4(4) - (-4)(0) = -16 \]

### Step 8: Combine the Results
Now, substituting back into the equation:
\[ V_A - V_B = - (6 - 16) = - (6 - 16) = - (-10) = -10 \, \text{V} \]

### Final Result
Thus, the potential difference between points A and B is:
\[ V_A - V_B = -10 \, \text{V} \]

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