Question

The electric field on a closed surface is due to all the charges in the universe, including the charges outside the closed surface. Explain why the total flux nevertheless proportional only to the charges that are inside the surface, with no apparent influence of the charges outside.

Short answer

The electric flux is proportional to inside charge of enclosed surface but not the outside charges because electric field lines are always normal to the enclosed surface for inside charges.

Step-by-step solution

Conceptual Explanation

The electric flux is the effect of the electric field lines passing through the certain cross section of an area. The effect varies with the orientation of cross section with electric field lines

Determination of Electric flux

The electric flux due to any charge or system of charges is given as:

$$\begin{gathered} \varphi =\overrightarrow{E}·\overrightarrow{A} \\ \varphi =EA\cos \theta \end{gathered}$$

Here, $E$ is the intensity of electric field lines due the charges, $A$ is the area of cross section and $\theta$ is the direction between electric field lines and area of cross section from where electric field lines passes.

The value of electric field lines varies with the direction between electric field lines and area of cross section. The flux is maximum for normal electric field lines to the surface and zero for the tangential electric field lines to the surface.

Reason for electric flux due to enclosed charge proportional to inside charge not for outside charges

The electric filed lines for inside charges of enclosed surface are normal to the surface so the electric flux is proportional to inside charges. The electric field lines for outside charges of enclosed surface are tangential to the surface so the electric flux has different values for different directions.

Therefore, the electric flux is proportional to inside charge of enclosed surface but not the outside charges because electric field lines are always normal to the enclosed surface for inside charges.