A circular coil of `'n'` turns is kept in a uniform magnetic fielf such that the plane of the coil is perpendicualr to the field. The magnetic flux associated with the coil is now `phi`. Now the coil is opened and made inot another circular coil od twice radius of the pervious coil and kept in the same field such that the plane of the coil is perpendicular to the field. The magnetic flux associated with this coil now is

Options

A. `phi`
B. `2 phi`
C. `(phi)/(4)`
D. `(phi)/(2)`

Explanation

To solve the problem, we need to analyze the situation step by step, focusing on the magnetic flux associated with the coils.

### Step 1: Determine the initial magnetic flux (φ) for the first coil
The magnetic flux (φ) through a coil is given by the formula:
\[ \phi = N \cdot B \cdot A \]
where:
- \(N\) = number of turns in the coil
- \(B\) = magnetic field strength
- \(A\) = area of the coil

For a circular coil of radius \(r\), the area \(A\) is:
\[ A = \pi r^2 \]
Thus, the initial magnetic flux can be expressed as:
\[ \phi = N \cdot B \cdot (\pi r^2) \]

### Step 2: Analyze the new coil created from the original coil
The coil is opened and reformed into a new coil with double the radius. The new radius is \(2r\).

### Step 3: Calculate the area of the new coil
The area \(A'\) of the new coil with radius \(2r\) is:
\[ A' = \pi (2r)^2 = \pi (4r^2) = 4\pi r^2 \]

### Step 4: Determine the number of turns in the new coil
Since the circumference of the coil remains constant, we can find the number of turns in the new coil. The original circumference was:
\[ C = N \cdot 2\pi r \]
The new circumference with radius \(2r\) is:
\[ C' = N' \cdot 2\pi (2r) = N' \cdot 4\pi r \]
Setting the two circumferences equal gives:
\[ N \cdot 2\pi r = N' \cdot 4\pi r \]
Cancelling \(2\pi r\) from both sides, we find:
\[ N' = \frac{N}{2} \]
So, the new coil has half the number of turns.

### Step 5: Calculate the new magnetic flux (φ')
Using the formula for magnetic flux again for the new coil:
\[ \phi' = N' \cdot B \cdot A' \]
Substituting \(N' = \frac{N}{2}\) and \(A' = 4\pi r^2\):
\[ \phi' = \left(\frac{N}{2}\right) \cdot B \cdot (4\pi r^2) \]
This simplifies to:
\[ \phi' = 2 \cdot N \cdot B \cdot (\pi r^2) \]
Recognizing that \(N \cdot B \cdot (\pi r^2) = \phi\), we have:
\[ \phi' = 2\phi \]

### Conclusion
Thus, the magnetic flux associated with the new coil is:
\[ \phi' = 2\phi \]