Question

If the earth electric flux entering and leaving an enclosed surface respectively are ${\phi _1}$ and ${\phi _2}$, then the electric charge inside the surface will be:
(A) $\dfrac{{{\phi _2} - {\phi _1}}}{{{\varepsilon _0 }}}$
(B) $\dfrac{{{\phi _2} + {\phi _1}}}{{{\varepsilon _0 }}}$
(C) $\dfrac{{{\phi _1} - {\phi _2}}}{{{\varepsilon _0 }}}$
(D) ${\varepsilon _0 }({\phi _2} - {\phi _1})$

Answer 1

To solve this question, we need to use the concept of Gauss Law for the electric flux. Using this concept, we can find the total charge by calculating the flux in the surface which will be a difference of the flux entering the surface and the flux leaving the surface.

Formula Used: The formula used in solving this question is given as
$\Rightarrow \oint {E.dl = \dfrac{{{Q_{enc}}}}{{{\varepsilon _0 }}}}$
Here, ${Q_{enc}}$ is the total charge enclosed within the surface, ${\varepsilon _0 }$ is the permittivity of free space, $E$ is the electric field and $\oint {E.dl}$ is the total flux in the surface.

Complete step by step answer
We know that Gauss law gives the total flux in the surface in terms of the total enclosed charge and permittivity of free space. Even though there are other ways of calculating total charge on the surface, since in the question we are given flux entering and leaving the surface so we should use Gauss Law to find the total enclosed charge.
Thus from Gauss Law, we know,
$\Rightarrow \oint {E.dl = \dfrac{{{Q_{enc}}}}{{{\varepsilon _0 }}}}$
But since, we are already given the flux entering and leaving the surface, we can calculate the total flux as,
$\Rightarrow \oint {E.dl = {\phi _2} - {\phi _1}}$
So, we can put this in the Gauss Law and get,
${\phi _2} - {\phi _1} = \dfrac{{{Q_{enc}}}}{{{\varepsilon _0 }}} \\\ \Rightarrow {\varepsilon _0 }({\phi _2} - {\phi _1}) = {Q_{enc}} \\\$
Hence, we get the Total enclosed charge within the surface as ${\varepsilon _0 }({\phi _2} - {\phi _1})$
$\therefore$Option (D) is correct out of the given options.

Note
Always remember that Gauss Law can be used only in certain conditions. We should either be given the flux entering and leaving the surface as in this question or the surface given to us should have spherical, cylindrical or planar symmetry. If the question does not belong to any of the above two categories, then we should use other methods like Coulomb’s Law to calculate the total charge.