The magnetic field in a certain region is given by `B = (4.0veci -1.8 veck) xx 10^-3 T`. How much flux passes through a `5.0 cm^2` area loop in this region if the loop lies flat on the xy-plane?

Options

A. `-0.9muWb`
B. `-1.0muWb`
C. -`0.03muWb`
D. `-0.04muWb`

Answer

To find the magnetic flux passing through a loop in a magnetic field, we can use the formula for magnetic flux, which is given by:

\[ \Phi = \mathbf{B} \cdot \mathbf{A} \]

where:
- \(\Phi\) is the magnetic flux,
- \(\mathbf{B}\) is the magnetic field vector,
- \(\mathbf{A}\) is the area vector of the loop.

### Step 1: Identify the magnetic field vector
The magnetic field is given as:
\[ \mathbf{B} = (4.0 \hat{i} - 1.8 \hat{k}) \times 10^{-3} \, \text{T} \]

### Step 2: Determine the area vector
The area of the loop is given as \(5.0 \, \text{cm}^2\). We need to convert this to square meters:
\[ \text{Area} = 5.0 \, \text{cm}^2 = 5.0 \times 10^{-4} \, \text{m}^2 \]

Since the loop lies flat on the xy-plane, the direction of the area vector \(\mathbf{A}\) will be along the z-axis:
\[ \mathbf{A} = 5.0 \times 10^{-4} \hat{k} \, \text{m}^2 \]

### Step 3: Calculate the dot product
Now we can calculate the magnetic flux using the dot product:
\[ \Phi = \mathbf{B} \cdot \mathbf{A} = (4.0 \hat{i} - 1.8 \hat{k}) \times 10^{-3} \cdot (5.0 \times 10^{-4} \hat{k}) \]

Calculating the dot product:
\[ \Phi = (4.0 \times 10^{-3} \hat{i} \cdot 5.0 \times 10^{-4} \hat{k}) + (-1.8 \times 10^{-3} \hat{k} \cdot 5.0 \times 10^{-4} \hat{k}) \]
Since \(\hat{i} \cdot \hat{k} = 0\) and \(\hat{k} \cdot \hat{k} = 1\):
\[ \Phi = 0 - (1.8 \times 5.0) \times 10^{-3} \times 10^{-4} \]
\[ \Phi = -9.0 \times 10^{-7} \, \text{Wb} \]

### Step 4: Convert to microWebers
To express the flux in microWebers:
\[ \Phi = -9.0 \times 10^{-7} \, \text{Wb} = -0.9 \, \mu\text{Wb} \]

### Conclusion
The magnetic flux passing through the loop is:
\[ \Phi = -0.9 \, \mu\text{Wb} \]