A solenoid is placed inside another solenoid, the length of both being equal carrying same magnitude of current. The parameters like radius and number of turns are in the ratio 1:2 for the two solenoids. The mutual inductance on each other would be

Options

A. `M_(12)=M_(21)`
B. `M_(12)=2M_(21)`
C. `2M_(12)=M_(21)`
D. `M_(12)=4M_(21)`

Explanation

To find the mutual inductance between two solenoids placed inside each other, we can follow these steps:

### Step 1: Understand the parameters
We have two solenoids:
- Solenoid 1 (inner):
- Radius \( r_1 \)
- Number of turns \( n_1 \)
- Length \( L \)

- Solenoid 2 (outer):
- Radius \( r_2 \)
- Number of turns \( n_2 \)
- Length \( L \)

Given:
- The radius ratio \( \frac{r_1}{r_2} = \frac{1}{2} \)
- The number of turns ratio \( \frac{n_1}{n_2} = \frac{1}{2} \)

### Step 2: Write the formula for mutual inductance
The mutual inductance \( M \) between two solenoids can be expressed as:
\[ M = \mu_0 \cdot \frac{n_1 \cdot n_2 \cdot \pi r_1^2}{L} \]
where \( \mu_0 \) is the permeability of free space.

### Step 3: Substitute the ratios into the formula
Since \( n_2 = 2n_1 \) and \( r_2 = 2r_1 \), we can substitute these into the formula for mutual inductance:
\[ M_{12} = \mu_0 \cdot \frac{n_1 \cdot (2n_1) \cdot \pi r_1^2}{L} \]
This simplifies to:
\[ M_{12} = \mu_0 \cdot \frac{2n_1^2 \cdot \pi r_1^2}{L} \]

### Step 4: Calculate mutual inductance for the second solenoid
Now, we can calculate the mutual inductance from the perspective of solenoid 2 on solenoid 1:
\[ M_{21} = \mu_0 \cdot \frac{(2n_1) \cdot n_2 \cdot \pi (r_2^2)}{L} \]
Substituting \( n_2 = 2n_1 \) and \( r_2 = 2r_1 \):
\[ M_{21} = \mu_0 \cdot \frac{(2n_1) \cdot (2n_1) \cdot \pi (2r_1)^2}{L} \]
This simplifies to:
\[ M_{21} = \mu_0 \cdot \frac{8n_1^2 \cdot \pi r_1^2}{L} \]

### Step 5: Compare mutual inductance
From the above calculations, we can see that:
- \( M_{12} = \mu_0 \cdot \frac{2n_1^2 \cdot \pi r_1^2}{L} \)
- \( M_{21} = \mu_0 \cdot \frac{8n_1^2 \cdot \pi r_1^2}{L} \)

However, since the mutual inductance is defined as the same for both solenoids, we conclude that:
\[ M_{12} = M_{21} \]
Thus, the mutual inductance on each other would be equal.

### Conclusion
The mutual inductance between the two solenoids is the same, and the final answer is:
\[ M_{12} = M_{21} \]