Question

Gauss’s law will be invalid if
A. There is magnetic monopoles
B. The inverse square law is not exactly true
C. The velocity of light is not universal constant
D. None of these

Answer 1

Hint: According to Gauss’s law, the total electric flux associated with the closed surface is equal to \[\dfrac{1}{{{\varepsilon _0}}}\] times the total charge enclosed by the surface. The total electric flux is the product of electric field E and area A of the surface. Assume that the inverse square law is not true and the electrostatic force is not inversely proportional to the square of the distance. The total electric flux associated with the closed surface does not depend on the shape of the surface.

Complete step by step answer:
We have according to Gauss’s law, the total electric flux associated with the closed surface is equal to \[\dfrac{1}{{{\varepsilon _0}}}\] times the total charge enclosed by the surface. Therefore, we can express Gauss’s law as,
\[{\phi _E} = \dfrac{{{q_{enc}}}}{{{\varepsilon _0}}}\]
Here, \[{\varepsilon _0}\] is the permittivity of the free space.

We know that the total electric flux is the product of electric field E and area A of the surface. Therefore,
\[E\,A = \dfrac{{{q_{enc}}}}{{{\varepsilon _0}}}\]

We have the expression for the electric force,
\[{F_e} = qE\]
\[ \Rightarrow E = \dfrac{{{F_e}}}{q}\]
Here, \[{F_e}\] is the electrostatic force. We have the expression for the electrostatic force,
\[{F_e} = k\dfrac{{{q^2}}}{{{r^2}}}\]
Here, k is the constant.

This law is also known as inverse square law. From the above equation, we have, the electrostatic force is inversely proportional to the square of the distance between the two charges. As given in the option C, if the inverse square law is not exactly true and the electrostatic force is inversely proportional to other than square of the distance, then the electric flux will be the function of distance r that is the electric flux will depend on the shape of the surface.

But from Gauss’s law, the electric flux associated with the surface is proportional to the total charge enclosed by the surface irrespective of its shape. Therefore, the option B is correct.

We know that magnetic monopoles never exist. Therefore, the option A is incorrect. We also know that the speed of light is always constant. Therefore, the option C is also incorrect.

So, the correct answer is option B.

Note:If we have the electrostatic force which is inversely proportional to the cube of the distance r, then the electric flux through the sphere of radius r will be,
\[\phi = EA = \left( {k\dfrac{q}{{{r^3}}}} \right)\left( {\pi {r^2}} \right)\]
\[ \Rightarrow \phi = k\pi \dfrac{q}{r}\]
Thus, the electric flux will be the function of distance r that it should not be for the validation of Gauss’s law.