Gauss's Law/Informal Explanation

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Gauss's Law Informal Explanation

Consider a point charge $q$ at the center of a sphere of radius $r$.

From Surface Area of Sphere, the area of the sphere is $4 \pi r^2$.

From Coulomb's Law of Electrostatics, the electric field at the surface of the sphere is of magnitude $\dfrac q {4 \pi \varepsilon_0 r^2}$.

Hence the product of the area and the magnitude of the electric field is:

$4 \pi r^2 \times \dfrac q {4 \pi \varepsilon_0 r^2} = \dfrac q {\varepsilon_0}$

Hence the total magnitude of the electric field is independent of the radius.


Analogy with Light

Consider a light bulb at the center of a spherical lampshade which is completely transparent.

All the light from the light bulb escapes.

That is, the total light emerging from the lampshade is independent of the radius of the lampshade.

In fact, the total light emerging from the lampshade is also independent of the shape of the lampshade.

All the light gets out, whether the lampshade is spherical or not.

Gauss's Law is an example of the same basic principle.


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