A short solenoid of radius a, number of turns per unit length `n_(1)`, and length L is kept coaxially inside a very long solenoid of radius b, number of turns per unit lengt `n_(2)`. What is the mutual inductance of the system

Options

A. `mu_(0)pib^(2)n_(1)n_(2)L`
B. `mu_(0)pia^(2)n_(1)n_(2)L^(2)`
C. `mu_(0)pia^(2)n_(1)n_(2)L`
D. `mu_(0)pib^(2)n_(1)n_(2)L^(2)`

Explanation

To find the mutual inductance \( M \) of a short solenoid placed inside a long solenoid, we can follow these steps:

### Step 1: Understand the Configuration
We have two solenoids:
- A short solenoid of radius \( a \), number of turns per unit length \( n_1 \), and length \( L \).
- A long solenoid of radius \( b \), number of turns per unit length \( n_2 \).

### Step 2: Area of the Short Solenoid
The area \( A \) of the cross-section of the short solenoid is given by:
\[ A = \pi a^2 \]

### Step 3: Magnetic Field Inside the Long Solenoid
The magnetic field \( B \) inside the long solenoid (which is uniform) is given by:
\[ B = \mu_0 n_2 I_2 \]
where \( I_2 \) is the current flowing through the long solenoid, and \( \mu_0 \) is the permeability of free space.

### Step 4: Magnetic Flux Through the Short Solenoid
The magnetic flux \( \Phi \) through the short solenoid due to the magnetic field of the long solenoid is:
\[ \Phi = B \cdot A = (\mu_0 n_2 I_2) \cdot (\pi a^2) = \mu_0 \pi a^2 n_2 I_2 \]

### Step 5: Induced EMF and Mutual Inductance
The mutual inductance \( M \) can be defined as the ratio of the magnetic flux linkage \( \Phi \) to the current \( I_2 \):
\[ M = \frac{\Phi}{I_2} = \frac{\mu_0 \pi a^2 n_2 I_2}{I_2} = \mu_0 \pi a^2 n_2 \]

### Step 6: Total Mutual Inductance Considering Short Solenoid
Since the short solenoid has \( n_1 \) turns per unit length, we need to consider the effect of \( n_1 \) on the mutual inductance:
\[ M = \mu_0 \pi a^2 n_1 n_2 L \]

### Final Expression for Mutual Inductance
Thus, the mutual inductance of the system is given by:
\[ M = \mu_0 \pi a^2 n_1 n_2 L \]

### Conclusion
The mutual inductance of the system is:
\[ M = \mu_0 \pi a^2 n_1 n_2 L \]