A solenoid of length `0.4 m` and diameter `0.6 m` consists of a single layer of `1000` turns of fine wire carrying a current of `5.0xx10^(-3)` ampere.The magnetic field strength on the axis at the middle is `pi/Nxx10^(-5)T`, then find value of `N`.

Answer

To solve the problem, we need to find the value of \( N \) given the parameters of a solenoid and the formula for the magnetic field strength at its center. Let's go through the steps systematically.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Length of the solenoid, \( L = 0.4 \, \text{m} \)
- Diameter of the solenoid, \( d = 0.6 \, \text{m} \)
- Number of turns, \( N = 1000 \)
- Current, \( I = 5.0 \times 10^{-3} \, \text{A} \)

2. **Calculate the Radius of the Solenoid:**
- The radius \( r \) is half of the diameter:
\[ r = \frac{d}{2} = \frac{0.6}{2} = 0.3 \, \text{m} \]

3. **Determine the Ratio of Length to Radius:**
- Calculate \( \frac{L}{r} \):
\[ \frac{L}{r} = \frac{0.4}{0.3} = \frac{4}{3} \]
- Since \( \frac{L}{r} > 1 \), we conclude that this is a long solenoid.

4. **Use the Formula for Magnetic Field Strength:**
- The magnetic field strength \( B \) at the center of a long solenoid is given by:
\[ B = \mu_0 n I \]
- Where \( n \) is the number of turns per unit length:
\[ n = \frac{N}{L} = \frac{1000}{0.4} = 2500 \, \text{turns/m} \]

5. **Substitute Values into the Magnetic Field Formula:**
- The permeability of free space \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \).
- Substitute \( n \) and \( I \) into the formula:
\[ B = (4\pi \times 10^{-7}) \times (2500) \times (5.0 \times 10^{-3}) \]

6. **Calculate the Magnetic Field Strength:**
- First, calculate \( 2500 \times 5.0 \times 10^{-3} \):
\[ 2500 \times 5.0 \times 10^{-3} = 12.5 \]
- Now, substitute back:
\[ B = 4\pi \times 10^{-7} \times 12.5 = 50\pi \times 10^{-7} \, \text{T} \]

7. **Express the Magnetic Field in the Required Form:**
- We want to express \( B \) in the form \( \frac{\pi}{N} \times 10^{-5} \):
\[ B = 50\pi \times 10^{-7} = \frac{\pi}{N} \times 10^{-5} \]

8. **Compare and Solve for \( N \):**
- Equating the two expressions:
\[ 50 \times 10^{-7} = \frac{1}{N} \times 10^{-5} \]
- Rearranging gives:
\[ N = \frac{1 \times 10^{-5}}{50 \times 10^{-7}} = \frac{10^{-5}}{50 \times 10^{-7}} = \frac{10^{-5}}{5 \times 10^{-6}} = 2 \]

### Final Answer:
The value of \( N \) is \( 2 \).