The displacement `bar(r )` of a charge Q in an electric field `E= e_(1) hat(i) + e_(2) hat(j) + e_(3) hat(k) " is " bar(r )= a hat(i) + b hat(j)`. The work done is

Options

A. `Q(a e_(1) + b e_(2))`
B. `Q sqrt((a e_(1))^(2) + (b e_(2))^(2))`
C. `Q (e_(1) + e_(2)) sqrt(a^(2) + b^(2))`
D. `(e_(1)^(2) + e_(2)^(2)) (a + b)`

Answer

To find the work done \( W \) when a charge \( Q \) is displaced in an electric field \( \mathbf{E} \), we can follow these steps:

### Step 1: Understand the formula for work done
The work done \( W \) by an electric field when a charge is moved is given by the formula:
\[ W = \mathbf{F} \cdot \mathbf{d} \]
where \( \mathbf{F} \) is the force acting on the charge and \( \mathbf{d} \) is the displacement vector.

### Step 2: Calculate the force on the charge
The force \( \mathbf{F} \) acting on the charge \( Q \) in an electric field \( \mathbf{E} \) is given by:
\[ \mathbf{F} = Q \mathbf{E} \]
Given that the electric field \( \mathbf{E} = e_1 \hat{i} + e_2 \hat{j} + e_3 \hat{k} \), we can express the force as:
\[ \mathbf{F} = Q (e_1 \hat{i} + e_2 \hat{j} + e_3 \hat{k}) \]

### Step 3: Write the displacement vector
The displacement vector \( \mathbf{d} \) is given as:
\[ \mathbf{d} = a \hat{i} + b \hat{j} \]

### Step 4: Substitute the values into the work done formula
Now substituting \( \mathbf{F} \) and \( \mathbf{d} \) into the work done formula:
\[ W = \mathbf{F} \cdot \mathbf{d} = (Q (e_1 \hat{i} + e_2 \hat{j} + e_3 \hat{k})) \cdot (a \hat{i} + b \hat{j}) \]

### Step 5: Calculate the dot product
Now, we calculate the dot product:
\[ W = Q \left( (e_1 \hat{i} + e_2 \hat{j} + e_3 \hat{k}) \cdot (a \hat{i} + b \hat{j}) \right) \]
Using the properties of the dot product:
\[ W = Q \left( e_1 a + e_2 b + e_3 \cdot 0 \right) \]
Since \( \hat{k} \) is orthogonal to both \( \hat{i} \) and \( \hat{j} \), the term involving \( e_3 \) becomes zero.

### Step 6: Final expression for work done
Thus, the work done is:
\[ W = Q (e_1 a + e_2 b) \]

### Final Answer
The work done \( W \) is given by:
\[ W = Q (e_1 a + e_2 b) \]