Correct Answer - Option 1 : 1 : 2

__CONCEPT:__

Gauss's law:

- According to Gauss law, the total electric flux linked with a closed surface called Gaussian surface is \(\frac{1}{ϵ_o}\) the charge enclosed by the closed surface.

\(\Rightarrow ϕ=\frac{Q}{ϵ_o}\)

Where ϕ = electric flux linked with a closed surface, Q = total charge enclosed in the surface, and ϵo = permittivity

Important points:

- Gauss’s law is true for any closed surface, no matter what its shape or size.
- The charges may be located anywhere inside the surface.

__CALCULATION:__

Given QA = q, and QB = 2q

Where AA and AB = surface area of the cube A and the cube B respectively

By the Gauss law, if the total charge enclosed in a closed surface is Q, then the total electric flux associated with it will be given as,

\(\Rightarrow ϕ=\frac{Q}{ϵ_o}\) -----(1)

By equation 1 the total flux associated with cube A is given as,

\(\Rightarrow ϕ_A=\frac{Q_A}{ϵ_o}\)

\(\Rightarrow ϕ_A=\frac{q}{ϵ_o}\) -----(2)

By equation 1 the total flux associated with cube B is given as,

\(\Rightarrow ϕ_B=\frac{Q_B}{ϵ_o}\)

\(\Rightarrow ϕ_B=\frac{2q}{ϵ_o}\) -----(3)

By equation 2 and equation 3,

\(\Rightarrow \frac{ϕ_A}{ϕ_B}=\frac{q}{ϵ_o}\times\frac{ϵ_o}{2q}\)

\(\Rightarrow \frac{ϕ_A}{ϕ_B}=\frac{1}{2}\)

- Hence, option 1 is correct.