Question

Electric field at a distance r from infinitely long conducting sheet is proportional to:
$\begin{align}
  & a){{r}^{-1}} \\
 & b){{r}^{-2}} \\
 & c){{r}^{\dfrac{3}{2}}} \\
 & d)none \\
\end{align}$

Answer 1

Hint: Let us find the electric field at a distance from an infinite conducting sheet in terms of uniform charge density. When we derive it in terms of uniform charge density, find out if there is any relation between the electric field and the distance between the point and the infinitely conducting sheet and derive the relation between them.
Formula used:
$E=\dfrac{\sigma }{2{{\varepsilon }_{0}}}$

Complete answer:
The field of a point charge, or a finite shaped charge, diverge as these proceed away from the charged object. This divergence in the field lines makes the field weaker as you move away from the charge. But once, the charged object is an infinite plane sheet, the field lines can't diverge, because if they do, they would be intersecting each other (A violation of the laws of physics). And since the field lines are parallel, they are uniform, which means, they would not depend on the distance from the plane. Let us try to formulate the electric field of an infinitely long conducting sheet at a distance using the charge density.
$E=\dfrac{\sigma }{2{{\varepsilon }_{0}}}$
If you observe this formula, the electric field does not depend on the distance at which it is calculated.

So, the correct answer is “Option D”.

Additional Information:
Gauss law relates the flux through any closed surface and the net charge enclosed within the surface. It states that the total flux of the electric field over any closed surface is equal to inverse epsilon times the net charge enclosed by the surface. This closed imaginary surface is called a gaussian surface. This law tells that the flux of electric particles through a closed surface depends only on the net charge inside the surface and not on the location of the charge outside the surface will not be considered because it does not contribute to the flux. Gauss law also known as gauss's flux theorem is a law relating the distribution of electric charge to the resulting electric field. It was first formulated by Joseph Louis Lagrange in 1773. They formulated this in the context of ellipsoids. This law is one of Maxwell's four equations.

Note:
In the case of an infinite plane, as you get closer, the contribution to the electric field becomes greater in the module. Direction of vectors become less perpendicular to the plane as you get closer, hence reducing the overall electric field in the perpendicular direction. Both these effects of each other subset the electric field is the same whatever the distance may be.