From the initial point A(1,2,0), to the final point D(3,2,0), a charge of 2C is moved through the points B (2,2,0) and C (3, 1, 0) along the path ABCD. The electric field in that region is 4 `bar(i)` N/C. The work done is given by

Options

A. `8J`
B. `-8J`
C. `16J`
D. `-4J`

Answer

To find the work done in moving a charge from point A to point D in an electric field, we can follow these steps:

### Step 1: Identify the Given Information
- Initial point A: \( (1, 2, 0) \)
- Final point D: \( (3, 2, 0) \)
- Charge \( Q = 2 \, \text{C} \)
- Electric field \( \vec{E} = 4 \hat{i} \, \text{N/C} \)

### Step 2: Calculate the Displacement Vector
The displacement vector \( \vec{dr} \) from point A to point D is given by:
\[ \vec{dr} = \vec{r_D} - \vec{r_A} \]
Where:
- \( \vec{r_A} = 1 \hat{i} + 2 \hat{j} + 0 \hat{k} \)
- \( \vec{r_D} = 3 \hat{i} + 2 \hat{j} + 0 \hat{k} \)

Calculating \( \vec{dr} \):
\[ \vec{dr} = (3 \hat{i} + 2 \hat{j}) - (1 \hat{i} + 2 \hat{j}) = (3 - 1) \hat{i} + (2 - 2) \hat{j} = 2 \hat{i} + 0 \hat{j} = 2 \hat{i} \]

### Step 3: Calculate the Force Acting on the Charge
The force \( \vec{F} \) acting on the charge in the electric field is given by:
\[ \vec{F} = Q \vec{E} \]
Substituting the values:
\[ \vec{F} = 2 \, \text{C} \cdot 4 \hat{i} \, \text{N/C} = 8 \hat{i} \, \text{N} \]

### Step 4: Calculate the Work Done
The work done \( W \) in moving the charge is given by the dot product of the force and the displacement:
\[ W = \vec{F} \cdot \vec{dr} \]
Substituting the values:
\[ W = (8 \hat{i}) \cdot (2 \hat{i}) = 8 \cdot 2 \cdot (\hat{i} \cdot \hat{i}) = 16 \, \text{J} \]

### Final Answer
The work done in moving the charge from point A to point D is \( 16 \, \text{J} \).

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