An infinite plane sheet of charge having uniform surface charge density $$+\sigma_s\ C/m^2$$ is placed on x−y plane. Another infinitely long line charge having uniform linear charge density $$+\lambda_e\ C/m$$ is placed at z = 4 m plane and parallel to y-axis. If the magnitude values $$|\sigma_s| = 2|\lambda_e|$$, then at point (0, 0, 2), the ratio of magnitudes of electric field values due to sheet charge to that of line charge is $$\pi\sqrt{n} : 1$$. The value of n is _____.

At point (0,0,2):

Electric field due to infinite sheet (on xy-plane with +σ_s): $$E_{sheet} = \frac{\sigma_s}{2\varepsilon_0}$$ (directed in +z direction at z=2)

Electric field due to line charge at z=4 parallel to y-axis: distance from line to point = $$\sqrt{0^2 + (4-2)^2} = 2$$ m

$$E_{line} = \frac{\lambda_e}{2\pi\varepsilon_0 \times 2} = \frac{\lambda_e}{4\pi\varepsilon_0}$$

Ratio: $$\frac{E_{sheet}}{E_{line}} = \frac{\sigma_s/(2\varepsilon_0)}{\lambda_e/(4\pi\varepsilon_0)} = \frac{2\pi\sigma_s}{\lambda_e}$$

Given $$|\sigma_s| = 2|\lambda_e|$$: $$\frac{E_{sheet}}{E_{line}} = \frac{2\pi \times 2\lambda_e}{\lambda_e} = 4\pi$$

We need $$4\pi = \pi\sqrt{n}$$, so $$\sqrt{n} = 4$$, $$n = 16$$.

The answer is 16.