A uniform electric field exists in `x-y` plane. The potential of points `A (-2m, 2m), B(-2m, 2m)` and `C(2m, 4m)` are `4V, 16 V` and 1`2 V` respectively. The electric field is

Options

A. `(4 hat(i)+5 hat(j))(V)/(m)`
B. `(3 hat(i)+4 hat(j))(V)/(m)`
C. `-(3hat(i)+4hat(j))(V)/(m)`
D. `(3hat(i)-4 hat(j))(V)/(m)`

Answer

To find the electric field in the given problem, we need to analyze the potential at the given points and use the relationship between electric field and electric potential.

### Step 1: Identify the points and their potentials
We have three points with their coordinates and potentials:
- Point A: \( A(-2 \, \text{m}, 2 \, \text{m}) \) with potential \( V_A = 4 \, \text{V} \)
- Point B: \( B(-2 \, \text{m}, 2 \, \text{m}) \) with potential \( V_B = 16 \, \text{V} \)
- Point C: \( C(2 \, \text{m}, 4 \, \text{m}) \) with potential \( V_C = 12 \, \text{V} \)

### Step 2: Calculate the electric field in the x-direction
The electric field \( E_x \) can be calculated using the formula:
\[ E_x = -\frac{\Delta V}{\Delta x} \]
where \( \Delta V \) is the change in potential and \( \Delta x \) is the change in x-coordinate.

Using points A and B:
- Since both A and B have the same x-coordinate (-2 m), we can use points A and C for the calculation.

\[ \Delta V = V_C - V_A = 12 \, \text{V} - 4 \, \text{V} = 8 \, \text{V} \]
\[ \Delta x = 2 \, \text{m} - (-2 \, \text{m}) = 4 \, \text{m} \]
Thus,
\[ E_x = -\frac{8 \, \text{V}}{4 \, \text{m}} = -2 \, \text{V/m} \]

### Step 3: Calculate the electric field in the y-direction
Now, we calculate the electric field \( E_y \) using the same formula:
\[ E_y = -\frac{\Delta V}{\Delta y} \]

Using points A and C:
\[ \Delta V = V_C - V_A = 12 \, \text{V} - 4 \, \text{V} = 8 \, \text{V} \]
\[ \Delta y = 4 \, \text{m} - 2 \, \text{m} = 2 \, \text{m} \]
Thus,
\[ E_y = -\frac{8 \, \text{V}}{2 \, \text{m}} = -4 \, \text{V/m} \]

### Step 4: Combine the electric field components
The total electric field \( \vec{E} \) can be expressed as:
\[ \vec{E} = E_x \hat{i} + E_y \hat{j} \]
Substituting the values we found:
\[ \vec{E} = (-2 \hat{i}) + (-4 \hat{j}) = -2 \hat{i} - 4 \hat{j} \]

### Final Answer
The electric field is:
\[ \vec{E} = -2 \hat{i} - 4 \hat{j} \, \text{V/m} \]