[Physics] Electric field due to a point charge


In my book it's written

Electric field due to a point charge is inversely proportional to $1/r^2$… This means that at all points on the spherical surface drawn around the point charge, the magnitude of $\vec E$ is same and doesn't depend on the direction of $\vec r$… Such a field is called spherically symmetric or radial field, i.e., a field which looks the same in all directions when seen from the point charge.

Now, why did they say magnitude of $\vec E$ doesn't depend on direction of $\vec r$? Also I will be glad if you could explain in simple English.

Picture of the full page

Best Answer

It's helpful that the posted image is sideways, because it already shows the answer: as you rotate the coordinate system, and advance by the same distance, $r$, everything looks exactly the same. This is true if you rotate in the plane of the table, or if you hold the page against a wall and rotate it.

If instead of rotating the coordinate system you were to imagine holding a smooth, uniformly colored sphere, like a ball bearing, you will find that no matter how you rotate it, it will appear the same.

This is spherical symmetry. Since everything is the same, the intensity calculated at one point will be the same as that for any other point. However, the radial vectors will all point in different directions - they all point directly away from the center, each one towards its own point on the surface.