A cubical surface surrounds a point charge q. Describe what happens to the total flux through the surface if (a) the charge is doubled, (b) the volume of the cube is doubled, (c) the surface is changed to a sphere, (d) the charge is moved to another location inside the surface, and (e) the charge is moved outside the surface.

a) flow must be doubled , b)   Ф₂ =∛2²  Ф₀, c)  Ф does not change , d)  flow is constant , e)  flow is constant

Explanation:

For this exercise let's use Gauss's law, which states that the flow is equal to the waxed load divided by the permissiveness of the vacuum

              Ф = E. dA = / ε₀

Let's apply this equation

a) if the charge inside the Gaussian surface is doubled by the equation the flow must be doubled

                Ф₀ =   / ε₀

                Ф₂ = 2   / ε₀ = 2 Ф₀

b) the volume is V₂ = 2V₀. Area is the volume divided by the length

              V = LLL = L³

              A = LL = L²

              A = V / L

              V₂ = 2V₀

              L³ = 2 L₀³

              L = ∛2  L₀

              V₂ = A₂ L

              A₂ = V₂ / L

              A₂ = 2V₀ / L = 2 L₀³ / (∛2 L₀)

              A₂ = 2 /∛2  V₀ / L₀

              A₂ = ∛2² A₀

              Ф₀ = E A₀

              Ф₂ = E A₂

              Ф₂ = E ∛2² A₀

               Ф₂ =∛2²  Ф₀

The flow doubles when the volume is increased by the factor

c) the flow does not depend on the shape of the surface, so Ф does not change

d) the flow does not depend on the position of the load, while inside the surface the flow is constant

e) when the load leaves the surface, the load inside drops to ero, so the flow changes to zero

            Ф = 0