Two solenoids A and B having radius of 5 cm and 3 cm respectively are placed coaxially such that A surrounds B. The length of both the solenoid is 30 cm and number of turns of A and B are 70 turns per cm and 50 turns per cm, respectively. What will be the mutual inductance of the system?

Answer

To find the mutual inductance of the two coaxial solenoids A and B, we can follow these steps:

### Step 1: Identify the given parameters
- Radius of solenoid A, \( R_A = 5 \, \text{cm} = 0.05 \, \text{m} \)
- Radius of solenoid B, \( R_B = 3 \, \text{cm} = 0.03 \, \text{m} \)
- Length of both solenoids, \( L = 30 \, \text{cm} = 0.3 \, \text{m} \)
- Number of turns per unit length of solenoid A, \( N_A = 70 \, \text{turns/cm} = 7000 \, \text{turns/m} \)
- Number of turns per unit length of solenoid B, \( N_B = 50 \, \text{turns/cm} = 5000 \, \text{turns/m} \)

### Step 2: Write the formula for mutual inductance
The mutual inductance \( M \) between two solenoids can be calculated using the formula:
\[ M = \mu_0 \cdot N_A \cdot N_B \cdot A_B \cdot L \]
where:
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{H/m} \) (permeability of free space)
- \( A_B \) is the cross-sectional area of solenoid B.

### Step 3: Calculate the cross-sectional area of solenoid B
The cross-sectional area \( A_B \) is given by:
\[ A_B = \pi R_B^2 \]
Substituting \( R_B = 0.03 \, \text{m} \):
\[ A_B = \pi (0.03)^2 = \pi \times 0.0009 \approx 0.002827 \, \text{m}^2 \]

### Step 4: Substitute values into the mutual inductance formula
Now substituting all the values into the mutual inductance formula:
\[ M = (4\pi \times 10^{-7}) \cdot (7000) \cdot (5000) \cdot (0.002827) \cdot (0.3) \]

### Step 5: Perform the calculations
Calculating the numerical values:
1. Calculate \( 7000 \cdot 5000 = 35,000,000 \)
2. Calculate \( 4\pi \times 10^{-7} \approx 1.256637 \times 10^{-6} \)
3. Now combine these:
\[ M \approx (1.256637 \times 10^{-6}) \cdot (35,000,000) \cdot (0.002827) \cdot (0.3) \]
Calculating this gives:
\[ M \approx 0.037 \, \text{H} \]
or converting to milliHenries:
\[ M \approx 37 \, \text{mH} \]

### Final Answer
The mutual inductance of the system is approximately \( 37 \, \text{mH} \).

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