Given:
- Ratio of hole diffusion current to electron diffusion current in an infinite p-n junction = 4
- Conductivity on p-side (σp) = 1.4 S/cm
- Conductivity on n-side (σn) = 2.8 S/cm
- We need to find the ratio of diffusion length Ln/Lp

Solution:

1. Introduction to p-n junction:
- A p-n junction is formed when a p-type semiconductor is brought into contact with an n-type semiconductor.
- The p-type region is dominated by holes (positive charge carriers), while the n-type region is dominated by electrons (negative charge carriers).
- At the junction, the holes and electrons diffuse across the junction due to the concentration gradient.

2. Diffusion current in a p-n junction:
- Diffusion current is the current due to the diffusion of charge carriers across the junction.
- The hole diffusion current (Jp) is given by Jp = qDp(dp/dx), where q is the charge of a hole, Dp is the hole diffusion coefficient, and dp/dx is the concentration gradient of holes.
- Similarly, the electron diffusion current (Jn) is given by Jn = qDn(dn/dx), where q is the charge of an electron, Dn is the electron diffusion coefficient, and dn/dx is the concentration gradient of electrons.

3. Ratio of hole diffusion current to electron diffusion current:
- The ratio of hole diffusion current to electron diffusion current is given by Jp/Jn = (qDp(dp/dx))/(qDn(dn/dx)).
- Since q is common to both terms, it cancels out.
- The ratio becomes Jp/Jn = (Dp(dp/dx))/(Dn(dn/dx)).

4. Relationship between diffusion length and diffusion coefficient:
- Diffusion coefficient is related to the diffusion length by D = μkT/q, where D is the diffusion coefficient, μ is the mobility, k is the Boltzmann constant, T is the temperature, and q is the charge.
- The diffusion length (L) is given by L = sqrt(Dτ), where L is the diffusion length, D is the diffusion coefficient, and τ is the carrier lifetime.

5. Calculation of Ln/Lp:
- Let Ln and Lp be the diffusion lengths of electrons and holes, respectively.
- The ratio of diffusion lengths Ln/Lp can be obtained by taking the square root of the ratio of diffusion coefficients (Dn/Dp).
- Since the conductivity (σ) is the reciprocal of the diffusion coefficient (σ = 1/D), we can write D = 1/σ.
- The ratio of diffusion lengths Ln/Lp is given by Ln/Lp = sqrt((Dn/σn)/(Dp/σp)).
- Substituting the given values, Ln/Lp = sqrt((2.8/1.4)/(1/2.8)) = sqrt((2.8^2)/(1.4^2)) = sqrt(7.84) = 2.8.

6. Conclusion:
- The ratio of diffusion length Ln/Lp is 2.8.
- The correct answer is not between 7.8 and 8.