# Electrostatic field-Field lines relationship [duplicate]

How is the $$\frac{1}{r^2}$$ dependence of the electric field intensity due to a stationary point charge consistent with the concept of field lines?

## marked as duplicate by Dvij Mankad, Emilio Pisanty, ZeroTheHero, John Rennie, stafusaFeb 24 at 15:35

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• Hi! Welcome to Physics Stack Exchange :) I am voting to close this question as of now as it is unclear what you are asking. Please elaborate on the specific question you want to ask. Thanks! :) – Dvij Mankad Feb 23 at 14:14
• – Emilio Pisanty Feb 23 at 14:52
• @EmilioPisanty No, that was not the answer I was looking for. Anyway, I worked it out yesterday using the concept of solid angles. I'll be posting my solution shortly. – Praveen K. Murugan Feb 24 at 12:04

## 2 Answers

The strength of an electric field is characterized by the density of electric field lines.

Notice that, if you draw "equally spaced" electric field lines all coming from a single point, they must spread out equally in all directions. But the surface area of the sphere around the origin point is $$4\pi r^2$$, so the density of the field lines at some distance $$r$$ from the charge, which is their constant number divided by the surface area of the sphere of radius $$r$$, goes as $$1/r^2$$, as predicted.

I've solved the problem using the concept of solid angle. In the figure, the charge q is at the origin and produces an electric field E in the surrounding space. To understand the dependence of the electric field lines on the area, or rather on the solid angle subtended by the area element, we must try to relate the area with the solid angle ∆Ω. We know that in a given solid angle the number of radial field lines is the same. For two points P1 and P2 at distances r1 and r2 from the charge, the element of area subtending the solid angle ∆Ω is ∆Ωr_1^2 at P1 and ∆Ωr_2^2 at P2 respectively. The number of lines (say n) cutting these area elements are the same. The number of field lines cutting, unit area element is therefore n/(ΔΩr_1^2 ) at P1 and n/(ΔΩr_2^2 ) at P2 respectively, since n and ΔΩ are common, the strength of the field clearly has a 1/r^2 dependence. Hence, the 1/r^2 dependence of the electric field due to a stationary point charge consistent with the concept of electric field lines