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. `2B_(0)L^(2)Wb`
C. `sqrt2B_(0)L^(2)Wb`
D. `4B_(0)L^(2)Wb`

Explanation

To solve the problem of calculating the magnetic flux through the loop ABCDEFA, we will follow these steps:

### Step 1: Identify the coordinates of the points
The loop is defined by the points:
- 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 area of the loop
The loop consists of two rectangular areas:
1. The area ABCD lies in the XY-plane.
2. The area AEF lies in the YZ-plane.

The dimensions of both rectangles are L x L.

- Area of rectangle ABCD (A1) = L * L = L²
- Area of rectangle AEF (A2) = L * L = L²

### Step 3: Define the area vectors
The area vector for each rectangle is defined as follows:
- For area ABCD (A1), the area vector is directed along the Z-axis:
\[ \vec{A_1} = L^2 \hat{k} \]

- For area AEF (A2), the area vector is directed along the Y-axis:
\[ \vec{A_2} = L^2 \hat{j} \]

### Step 4: Define the magnetic field
The magnetic field is given as:
\[ \vec{B} = B_0 (\hat{i} + \hat{k}) \]

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

**For area ABCD (A1):**
\[ \Phi_1 = \vec{B} \cdot \vec{A_1} = B_0 (\hat{i} + \hat{k}) \cdot (L^2 \hat{k}) = B_0 L^2 (\hat{k} \cdot \hat{k}) = B_0 L^2 \]

**For area AEF (A2):**
\[ \Phi_2 = \vec{B} \cdot \vec{A_2} = B_0 (\hat{i} + \hat{k}) \cdot (L^2 \hat{j}) = B_0 L^2 (\hat{i} \cdot \hat{j} + \hat{k} \cdot \hat{j}) = 0 \]
(Since \(\hat{i} \cdot \hat{j} = 0\) and \(\hat{k} \cdot \hat{j} = 0\))

### Step 6: Calculate the total magnetic flux
The total magnetic flux through the loop ABCDEFA is the sum of the fluxes through the two areas:
\[ \Phi_{total} = \Phi_1 + \Phi_2 = B_0 L^2 + 0 = B_0 L^2 \]

### Step 7: Final answer
The total magnetic flux passing through the loop ABCDEFA is:
\[ \Phi_{total} = B_0 L^2 \]