The magnetic field in a certain region is given by
`vecB=(4hati-hatk)` tesla. How much magnetic flux passes through the loop of area 100.0cm2 lies flat in xy plane ?

Options

A. `-0.01 `
B. `-0.02`
C. `-0.03`
D. `-0.04`

Explanation

To solve the problem of finding the magnetic flux through a loop in the xy-plane with a given magnetic field, we can follow these steps:

### Step 1: Understand the magnetic field and area vector
The magnetic field is given as:
\[ \vec{B} = (4\hat{i} - 4\hat{k}) \, \text{T} \]
The area of the loop is given as \(100.0 \, \text{cm}^2\). First, we need to convert this area into square meters:
\[ 100.0 \, \text{cm}^2 = 100.0 \times 10^{-4} \, \text{m}^2 = 0.01 \, \text{m}^2 \]

Since the loop lies flat in the xy-plane, the area vector \(\vec{A}\) will be perpendicular to the surface of the loop, which points in the positive z-direction. Therefore, the area vector can be expressed as:
\[ \vec{A} = 0.01 \hat{k} \, \text{m}^2 \]

### Step 2: Calculate the magnetic flux
The magnetic flux \(\Phi\) through the loop is given by the dot product of the magnetic field vector \(\vec{B}\) and the area vector \(\vec{A}\):
\[ \Phi = \vec{B} \cdot \vec{A} \]
Substituting the values we have:
\[ \Phi = (4\hat{i} - 4\hat{k}) \cdot (0.01 \hat{k}) \]

### Step 3: Perform the dot product
The dot product can be computed as follows:
\[ \Phi = 4\hat{i} \cdot (0.01 \hat{k}) + (-4\hat{k}) \cdot (0.01 \hat{k}) \]
Since \(\hat{i} \cdot \hat{k} = 0\) (they are perpendicular), the first term becomes zero:
\[ \Phi = 0 + (-4)(0.01) = -0.04 \, \text{Wb} \]

### Step 4: Final answer
Thus, the magnetic flux passing through the loop is:
\[ \Phi = -0.04 \, \text{Wb} \]