A loop made of straight edegs has six corners at `A(0,0,0), B(L, O,0) C(L,L,0), D(0,L,0) E(0,L,L)` and `F(0,0,L)`. Where `L` is in meter. A magnetic field `B = B_(0)(hat(i) + hat(k))T` is present in the region. The flux passing through the loop `ABCDEFA` (in that order) is

Options

A. `B_(0)L^(2)Wb`
B. `2 B_(0) L^(2) Wb`.
C. `2 B_(0) L^(2) Wb`.
D. `4 B_(0) L^(2) Wb`.

Explanation

To find the magnetic flux passing through the loop \( ABCDEFA \), we can follow these steps:

### Step 1: Identify the Coordinates of the Points
The coordinates of the points are given as:
- \( A(0, 0, 0) \)
- \( B(L, 0, 0) \)
- \( C(L, L, 0) \)
- \( D(0, L, 0) \)
- \( E(0, L, L) \)
- \( F(0, 0, L) \)

### Step 2: Determine the Magnetic Field
The magnetic field is given as:
\[ \mathbf{B} = B_0 (\hat{i} + \hat{k}) \, \text{T} \]

### Step 3: Calculate the Area of the Loop
The loop consists of two rectangular areas:
1. Area \( ABCD \) lies in the \( xy \)-plane.
2. Area \( AFE \) lies in the \( yz \)-plane.

#### Area \( ABCD \):
- The area vector \( \mathbf{A_1} \) for the rectangle \( ABCD \) is perpendicular to the \( xy \)-plane and points in the \( z \)-direction:
\[ \mathbf{A_1} = L^2 \hat{k} \]

#### Area \( AFE \):
- The area vector \( \mathbf{A_2} \) for the rectangle \( AFE \) is perpendicular to the \( yz \)-plane and points in the \( x \)-direction:
\[ \mathbf{A_2} = L^2 \hat{i} \]

### Step 4: Total Area Vector
The total area vector \( \mathbf{A} \) is the sum of the two area vectors:
\[ \mathbf{A} = \mathbf{A_1} + \mathbf{A_2} = L^2 \hat{k} + L^2 \hat{i} = L^2 (\hat{i} + \hat{k}) \]

### Step 5: Calculate the Magnetic Flux
The magnetic flux \( \Phi \) through the loop is given by the dot product of the magnetic field \( \mathbf{B} \) and the area vector \( \mathbf{A} \):
\[ \Phi = \mathbf{B} \cdot \mathbf{A} \]

Substituting the values:
\[ \Phi = B_0 (\hat{i} + \hat{k}) \cdot L^2 (\hat{i} + \hat{k}) \]

### Step 6: Perform the Dot Product
Calculating the dot product:
\[ \Phi = B_0 L^2 \left( \hat{i} \cdot \hat{i} + \hat{i} \cdot \hat{k} + \hat{k} \cdot \hat{i} + \hat{k} \cdot \hat{k} \right) \]
\[ = B_0 L^2 (1 + 0 + 0 + 1) = B_0 L^2 \cdot 2 \]

### Final Answer
Thus, the magnetic flux through the loop is:
\[ \Phi = 2 B_0 L^2 \, \text{Wb} \]