Introduction:
In this problem, we are given a short solenoid with a certain number of turns per unit length. We are asked to find the magnetic field at a specific point on the axis of the solenoid.

Given:
- Number of turns per unit length: n
- Diameter of the solenoid: D
- Length of the solenoid: L = 8D/15
- Current flowing through the solenoid: I
- Distance of point P from the near end of the solenoid: H = 2D/3

Approach:
To find the magnetic field at point P, we can use the formula for the magnetic field inside a solenoid, which is given by:

B = μ₀ * n * I

where:
- B is the magnetic field
- μ₀ is the permeability of free space (a constant)
- n is the number of turns per unit length
- I is the current flowing through the solenoid

However, since point P is not at the center of the solenoid, we need to take into account the effect of the non-uniform magnetic field. We can consider the solenoid as a combination of smaller loops, each with a different radius and carrying a different current.

Calculation:
To calculate the magnetic field at point P, we can divide the solenoid into smaller loops and sum up the contributions from each loop. Let's consider a small section of the solenoid of length dx at a distance x from the near end. The current flowing through this section is given by:

dI = I * (dx/L)

The magnetic field produced by this small section at point P is given by:

dB = μ₀ * dI * r² / (2 * (r² + H²)³/2)

where r is the distance from the small section to point P.

We can integrate this expression over the length of the solenoid from x = 0 to x = L to find the total magnetic field at point P:

B = ∫ dB = ∫ μ₀ * I * r² / (2 * (r² + H²)³/2) * (dx/L) = μ₀ * n * I * ∫ r² / (2 * (r² + H²)³/2) * dx

By solving this integral, we can find the magnetic field at point P.