Correct Answer - Option 1 :

\(\frac{3q}{\epsilon_o}\)
__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.

\(⇒ ϕ=\frac{Q}{ϵ_o}\)

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

__CALCULATION:__

Given Q1 = -3q, Q2 = +2q, and Q3 = +4q

The net charge enclosed in the sphere is given as,

⇒ Q = Q1 + Q2 + Q3

⇒ Q = -3q + 2q + 4q

⇒ Q = 3q

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,

\(⇒ ϕ=\frac{Q}{ϵ_o}\) -----(1)

By equation 1 the total flux linked with the sphere is given as,

\(⇒ ϕ=\frac{Q}{ϵ_o}\)

\(⇒ ϕ=\frac{3q}{ϵ_o}\)

- Hence, option 1 is correct.

- 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.