An electric field given by `E=2E_(0)veci+3E_(0) vecj-5E_(0)veck` exists in a cerrtain region, where `E_(0)=100N//C`. How much flux will pass through a rectangular surface of area 0.2 `m^(2)`, placed in this region, in such away that it is parallel to Y-axis?

Options

A. `20Nm^(2)//C`
B. `40Nm^(2)//C`
C. `60Nm^(2)//C`
D. `80Nm^(2)//C`

Explanation

To solve the problem of calculating the electric flux through a rectangular surface in a given electric field, we can follow these steps:

### Step 1: Identify the Electric Field and Area Vector
The electric field is given as:
\[ \mathbf{E} = 2E_0 \hat{i} + 3E_0 \hat{j} - 5E_0 \hat{k} \]
where \(E_0 = 100 \, \text{N/C}\).

The area of the rectangular surface is given as \(A = 0.2 \, \text{m}^2\) and it is parallel to the Y-axis. Therefore, the area vector \(\mathbf{A}\) can be expressed as:
\[ \mathbf{A} = A \hat{j} = 0.2 \hat{j} \, \text{m}^2 \]

### Step 2: Calculate the Dot Product
The electric flux \(\Phi\) through the surface is given by the formula:
\[ \Phi = \mathbf{E} \cdot \mathbf{A} \]
To calculate this, we need to find the dot product of \(\mathbf{E}\) and \(\mathbf{A}\):
\[ \Phi = (2E_0 \hat{i} + 3E_0 \hat{j} - 5E_0 \hat{k}) \cdot (0.2 \hat{j}) \]

### Step 3: Compute the Dot Product
Using the properties of dot products, we have:
\[ \Phi = 2E_0 \hat{i} \cdot (0.2 \hat{j}) + 3E_0 \hat{j} \cdot (0.2 \hat{j}) - 5E_0 \hat{k} \cdot (0.2 \hat{j}) \]
Since the dot product of perpendicular vectors is zero:
- \(\hat{i} \cdot \hat{j} = 0\)
- \(\hat{k} \cdot \hat{j} = 0\)

Thus, we only need to consider the second term:
\[ \Phi = 3E_0 \cdot 0.2 \]

### Step 4: Substitute the Value of \(E_0\)
Now, substituting \(E_0 = 100 \, \text{N/C}\):
\[ \Phi = 3 \cdot 100 \cdot 0.2 = 60 \, \text{N m}^2/\text{C} \]

### Final Answer
The electric flux through the surface is:
\[ \Phi = 60 \, \text{N m}^2/\text{C} \]

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