Question

Charge ${{Q}_{1}}$ and ${{Q}_{2}}$ lie inside and outside respectively of a closed surface S. Let E be the field at any point on S and $\phi $ be the flux of E over S. Choose the incorrect statement.
A. If ${{Q}_{1}}$ changes, both E and $\phi $ will change.
B. If ${{Q}_{2}}$ changes, E will change but $\phi $will not change.
C. If ${{Q}_{1}}=0$ and ${{Q}_{2}}\ne 0$then $E\ne 0$ but $\phi =0$.
D. If ${{Q}_{1}}\ne 0$ and ${{Q}_{2}}=0$then $E=0$ but $\phi \ne 0$

Answer 1

Hint: As a first step, one could read the question well. Then you could recall as to how we find the electric field at a particular point and also the electric flux at that point. After that you could get help from Gauss's law to arrive at a conclusion for each of the given statements.
Formula used:
Gauss’s law,
$\phi =\dfrac{{{q}_{enclosed}}}{{{\varepsilon }_{0}}}$

Complete step-by-step solution:
In the question, we are given two charges located inside and outside of a closed sphere S. We are also said that E is the electric field on point S and $\phi $ is the flux of E on S. We are supposed to find the correct statements among the given ones.

The point to be kept in mind is that, electric field at any point would be due to all of the charges in the point’s vicinity be it outside or inside. Now for the electric flux at a point, the charge enclosed by the closed surface would solely be responsible for it. From Gauss’s law, we know that, electric flux,
$\phi =\dfrac{{{q}_{enclosed}}}{{{\varepsilon }_{0}}}$
Here, the charge enclosed by the closed surface is ${{Q}_{1}}$.
For option A, it is true as the change in ${{Q}_{1}}$ would cause change in the values of E as well as $\phi $.
For option B, it is also true as ${{Q}_{2}}$ is not enclosed by the surface and wouldn't affect the flux but only the electric field.
For the option C, it is also true, as the enclosed charge being zero will only cause the flux to be zero. However the electric field would still not be zero as there is another charge present.
Option D is a false statement. Electric field wouldn’t be zero as there would be the field due to charge ${{Q}_{1}}$ at the point.
Hence, option D is the incorrect statement.

Note: For any questions involving finding the false statement from the given ones, one should deal with it in a step by step manner. You should consider each statement one by one and justify whether they are true or false. Thus we could arrive at a conclusion avoiding further confusions.