A coil of area `5cm^(2)` and having 20 turns is placed in a uniform magnetic field of `10^(3)` gauss. The normal to the plane of the coil makes an angle of `60^(@)` with the magnetic field. The flux in Maxwell through the coil is:

Options

A. `10^(5)`
B. `5xx10^(4)`
C. `2xx10^(4)`
D. `5xx10^(3)`

Explanation

To solve the problem of finding the magnetic flux through a coil placed in a uniform magnetic field, we can follow these steps:

### Step 1: Convert the area from cm² to m²
The area of the coil is given as \(5 \, \text{cm}^2\). We need to convert this to square meters.

\[ \text{Area} = 5 \, \text{cm}^2 = 5 \times 10^{-4} \, \text{m}^2 \]

### Step 2: Convert the magnetic field from gauss to tesla
The magnetic field is given as \(10^3 \, \text{gauss}\). We need to convert this to tesla using the conversion \(1 \, \text{gauss} = 10^{-4} \, \text{tesla}\).

\[ \text{Magnetic Field} = 10^3 \, \text{gauss} = 10^3 \times 10^{-4} \, \text{T} = 0.1 \, \text{T} \]

### Step 3: Identify the number of turns and the angle
The number of turns \(N\) is given as \(20\), and the angle \(\theta\) between the normal to the coil and the magnetic field is \(60^\circ\).

### Step 4: Use the formula for magnetic flux
The magnetic flux \(\Phi\) through the coil is given by the formula:

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

Where:
- \(N\) = number of turns = 20
- \(B\) = magnetic field in tesla = 0.1 T
- \(A\) = area in m² = \(5 \times 10^{-4} \, \text{m}^2\)
- \(\theta\) = angle between the magnetic field and the normal to the coil = \(60^\circ\)

### Step 5: Calculate \(\cos(60^\circ)\)
\[ \cos(60^\circ) = \frac{1}{2} \]

### Step 6: Substitute the values into the formula
Now we can substitute all the values into the formula for magnetic flux:

\[ \Phi = 20 \cdot 0.1 \cdot (5 \times 10^{-4}) \cdot \cos(60^\circ) \]

\[ \Phi = 20 \cdot 0.1 \cdot (5 \times 10^{-4}) \cdot \frac{1}{2} \]

### Step 7: Simplify the expression
\[ \Phi = 20 \cdot 0.1 \cdot (5 \times 10^{-4}) \cdot 0.5 \]

\[ \Phi = 20 \cdot 0.1 \cdot 2.5 \times 10^{-4} \]

\[ \Phi = 20 \cdot 0.25 \times 10^{-4} \]

\[ \Phi = 5 \times 10^{-4} \, \text{Wb} \]

### Step 8: Convert the flux to maxwell
Since \(1 \, \text{Wb} = 10^8 \, \text{Maxwell}\):

\[ \Phi = 5 \times 10^{-4} \, \text{Wb} \times 10^8 \, \text{Maxwell/Wb} = 5 \times 10^4 \, \text{Maxwell} \]

### Final Answer
The magnetic flux through the coil is:

\[ \Phi = 5 \times 10^4 \, \text{Maxwell} \]

---