On making a coil of copper wire of length  and coil radius r, the value of self inductance is obtained as L. If the coil of same wire, but of coil radius r/2, is made, the value of self inductance will be-

Options

A. 2L
B. L
C. 4L
D. `L//2`

Answer

To solve the problem of finding the self-inductance of a coil made from the same copper wire but with a different radius, we will follow these steps:

### Step 1: Understand the formula for self-inductance
The self-inductance \( L \) of a coil is given by the formula:

\[ L = \frac{\mu_0 \pi N^2 r^2}{l} \]

where:
- \( \mu_0 \) is the permeability of free space (a constant),
- \( N \) is the number of turns in the coil,
- \( r \) is the radius of the coil,
- \( l \) is the length of the coil.

### Step 2: Determine the number of turns \( N \)
The number of turns \( N \) in the coil can be calculated as:

\[ N = \frac{L}{2\pi r} \]

where \( L \) is the length of the wire used to make the coil.

### Step 3: Calculate self-inductance for the original coil
For the original coil with radius \( r \):

\[ L = \frac{\mu_0 \pi N^2 r^2}{l} \]

### Step 4: Calculate self-inductance for the new coil
Now, we consider the new coil with radius \( \frac{r}{2} \). The new number of turns \( N' \) can be calculated as:

\[ N' = \frac{L}{2\pi \left(\frac{r}{2}\right)} = \frac{L}{\pi r} = 2N \]

This shows that the number of turns doubles when the radius is halved.

### Step 5: Substitute \( N' \) and \( r' \) into the self-inductance formula
Now, substituting \( N' \) and \( r' = \frac{r}{2} \) into the self-inductance formula:

\[ L' = \frac{\mu_0 \pi (N')^2 \left(\frac{r}{2}\right)^2}{l} \]

Substituting \( N' = 2N \):

\[ L' = \frac{\mu_0 \pi (2N)^2 \left(\frac{r}{2}\right)^2}{l} \]

### Step 6: Simplify the expression
Calculating \( L' \):

\[ L' = \frac{\mu_0 \pi (4N^2) \left(\frac{r^2}{4}\right)}{l} \]

This simplifies to:

\[ L' = \frac{\mu_0 \pi N^2 r^2}{l} = L \]

### Step 7: Conclusion
Thus, the self-inductance of the new coil with radius \( \frac{r}{2} \) is:

\[ L' = L \]

### Final Answer
The value of self-inductance for the coil with radius \( \frac{r}{2} \) is \( L \).

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