The magnetic field in a certain region is given by
`vecB=(4hati+2hatj-3hatk)` tesla. How much magnetic flux passes through the loop of area `0.2 m^2` in this region if the loop lies flat in xy plane ?

Options

A. `-0.1`
B. `-0.2`
C. `0.4`
D. `-0.6`

Explanation

To find the magnetic flux passing through the loop, we can follow these steps:

### Step 1: Understand the Magnetic Field and Area Vector
The magnetic field is given by:
\[ \vec{B} = 4\hat{i} + 2\hat{j} - 3\hat{k} \text{ tesla} \]
The loop lies flat in the xy-plane. Therefore, the area vector \(\vec{A}\) of the loop is perpendicular to the plane of the loop, which means it points in the z-direction. The area of the loop is given as \(0.2 \, m^2\).

### Step 2: Define the Area Vector
Since the loop is in the xy-plane, the area vector can be expressed as:
\[ \vec{A} = 0.2 \hat{k} \text{ m}^2 \]

### Step 3: 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:
\[ \Phi = (4\hat{i} + 2\hat{j} - 3\hat{k}) \cdot (0.2\hat{k}) \]

### Step 4: Compute the Dot Product
Now, we compute the dot product:
\[ \Phi = 4\hat{i} \cdot (0.2\hat{k}) + 2\hat{j} \cdot (0.2\hat{k}) - 3\hat{k} \cdot (0.2\hat{k}) \]
Since \(\hat{i} \cdot \hat{k} = 0\) and \(\hat{j} \cdot \hat{k} = 0\), the first two terms are zero:
\[ \Phi = 0 + 0 - 3 \cdot 0.2 \]
\[ \Phi = -0.6 \text{ Wb} \]

### Step 5: Final Result
Thus, the magnetic flux passing through the loop is:
\[ \Phi = -0.6 \text{ Wb} \]