Consider a loop ABCDEFA. With coordinates A (0, 0, 0), B(5, 0, 0), C(5, 5, 0), D(0, 5, 0) E(0, 5, 5) and F(0, 0, 5). Find magnetic flux through loop due to magnetic field ` vecB = 3 hati+4 hatk`

Answer

To find the magnetic flux through the loop ABCDEFA due to the magnetic field \( \vec{B} = 3 \hat{i} + 4 \hat{k} \), we will follow these steps:

### Step 1: Identify the coordinates of the loop
The coordinates of the loop ABCDEFA are:
- A (0, 0, 0)
- B (5, 0, 0)
- C (5, 5, 0)
- D (0, 5, 0)
- E (0, 5, 5)
- F (0, 0, 5)

### Step 2: Determine the area of the loop
The loop ABCD lies in the xy-plane, forming a square. The side length of the square is 5 units (from A to B or A to D).

The area \( A \) of the square is given by:
\[ A = \text{side}^2 = 5^2 = 25 \, \text{square units} \]

### Step 3: Determine the area vector
The area vector \( \vec{A} \) for the loop ABCD is perpendicular to the plane of the loop. Since the loop lies in the xy-plane, the area vector points in the positive z-direction:
\[ \vec{A}_{ABCD} = 25 \hat{k} \]

For the loop ADEF, which is vertical, the area vector points in the negative x-direction:
\[ \vec{A}_{ADEF} = 25 \hat{i} \]

### Step 4: Combine the area vectors
The total area vector \( \vec{A} \) for the entire loop ABCDEFA is the sum of the area vectors of the two sections:
\[ \vec{A} = \vec{A}_{ABCD} + \vec{A}_{ADEF} = 25 \hat{k} + 25 \hat{i} \]

### Step 5: Calculate the magnetic flux
The magnetic flux \( \Phi \) through the loop is given by the dot product of the magnetic field \( \vec{B} \) and the area vector \( \vec{A} \):
\[ \Phi = \vec{B} \cdot \vec{A} \]
Substituting the values:
\[ \vec{B} = 3 \hat{i} + 4 \hat{k} \]
\[ \vec{A} = 25 \hat{i} + 25 \hat{k} \]

Calculating the dot product:
\[ \Phi = (3 \hat{i} + 4 \hat{k}) \cdot (25 \hat{i} + 25 \hat{k}) = (3 \cdot 25) + (4 \cdot 25) = 75 + 100 = 175 \, \text{Weber} \]

### Final Answer
The magnetic flux through the loop ABCDEFA is \( 175 \, \text{Weber} \).

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