A circular loop of radius 10 cm is placed in a region of magnetic field of 0.5 T with its plane parallel to the magnetic field. Calculate the magnetic flux through the coil.

Answer

To calculate the magnetic flux through a circular loop placed in a magnetic field, we can use the formula for magnetic flux (Φ):

\[ \Phi = B \cdot A \cdot \cos(\theta) \]

Where:
- \( \Phi \) is the magnetic flux,
- \( B \) is the magnetic field strength,
- \( A \) is the area of the loop,
- \( \theta \) is the angle between the magnetic field and the normal (perpendicular) to the surface of the loop.

### Step 1: Identify the parameters
- The radius of the circular loop \( r = 10 \, \text{cm} = 0.1 \, \text{m} \) (convert cm to m).
- The magnetic field strength \( B = 0.5 \, \text{T} \).
- The angle \( \theta \) between the magnetic field and the area vector of the loop is \( 90^\circ \) since the plane of the loop is parallel to the magnetic field.

### Step 2: Calculate the area of the loop
The area \( A \) of a circular loop is given by the formula:

\[ A = \pi r^2 \]

Substituting the radius:

\[ A = \pi (0.1)^2 = \pi (0.01) \approx 0.0314 \, \text{m}^2 \]

### Step 3: Calculate the magnetic flux
Now, substituting the values into the magnetic flux formula:

\[ \Phi = B \cdot A \cdot \cos(\theta) \]

Since \( \theta = 90^\circ \):

\[ \cos(90^\circ) = 0 \]

Thus,

\[ \Phi = 0.5 \cdot 0.0314 \cdot 0 = 0 \]

### Conclusion
The magnetic flux through the coil is:

\[ \Phi = 0 \, \text{Wb} \, (\text{Weber}) \]