Two insulated wires are wound on the same hollow cylinder, s as to from two solenoids shering a common air-filled core. Let `l` be the length of the core, A the cross -sectional area of the core, `N_(1)` the number of times the first wire is wound around the core, and `N_(2)` the number of times the second wire is wound around the core. Find the mutual inductance of the two solenoids, neglecting the end effects.

Explanation

To find the mutual inductance \( M \) of the two solenoids sharing a common air-filled core, we can follow these steps:

### Step 1: Understand the Magnetic Field Created by the First Solenoid
When a current \( I_1 \) flows through the first solenoid (with \( N_1 \) turns), it generates a magnetic field \( B_1 \) in the core. The formula for the magnetic field inside a solenoid is given by:

\[ B_1 = \frac{\mu_0 N_1 I_1}{l} \]
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