A system consists of two parallel plane carrying currents producing a uniform magnetic field of induction B between the planes. Outside this space there is no magnetic field. The magnetic force acting per unit area of each plane is found to be `B^2//Nmu_0`. Find N.

Explanation

To solve the problem, we need to find the value of \( N \) given that the magnetic force acting per unit area of each plane is \( \frac{B^2}{N \mu_0} \).

### Step-by-step Solution:

1. **Understanding the Magnetic Field**:
- We have two parallel planes carrying currents that produce a uniform magnetic field \( B \) between them.
- The magnetic field due to each plane is \( \frac{B}{2} \) because the total field \( B \) is the sum of the fields from both planes.

2. **Magnetic Force on Each Plane**:
- The magnetic force \( F \) acting on one of the planes can be expressed as:
\[ F = B \cdot I \cdot L \]
where \( I \) is the current and \( L \) is the length of the wire (or the effective length of the plane).

3. **Calculating Force per Unit Area**:
- The force per unit area \( f \) can be calculated as:
\[ f = \frac{F}{A} \]
where \( A \) is the area of the plane. If we consider the area as \( L \times w \) (where \( w \) is the width), we can express the force per unit area in terms of the magnetic field.

4. **Using the Relation for Current Density**:
- The current density \( J \) can be related to the magnetic field \( B \) using the formula:
\[ J = \frac{B}{\mu_0} \]
- Therefore, substituting \( J \) into the force equation, we have:
\[ f = \frac{B^2}{2 \mu_0} \]
- This is because the force per unit area can also be expressed as \( f = \frac{B^2}{\mu_0} \) for two parallel planes.

5. **Comparing with Given Expression**:
- We know from the problem that the force per unit area is given by:
\[ f = \frac{B^2}{N \mu_0} \]
- Setting the two expressions for \( f \) equal gives:
\[ \frac{B^2}{2 \mu_0} = \frac{B^2}{N \mu_0} \]

6. **Solving for \( N \)**:
- By canceling \( B^2 \) and \( \mu_0 \) from both sides, we find:
\[ N = 2 \]

### Final Answer:
Thus, the value of \( N \) is \( 2 \).