Gaussian theorem

Question

Using the Gaussian theorem, prove the distance dependence of the point charge's electric field at 1 / r^2!

Answer 1

Hello Emma!

Gauss's Law states that the surface integral of E • dA = Qenc/.

Now let's imagine a point charge, sitting at the center of a sphere.

We need to figure out the surface integral of E • dA. dA simply means a tiny, tiny area of the sphere we are imagining. Taking a surface integral means to sum up all these tiny areas on the surface of the sphere. What is the formula for the surface area (the sum of all the tiny areas) of a sphere? That's 4 * pi * r^2. So, the surface integral of E • dA evaluates to E(r) * 4 * pi * r^2. (I say E(r) just to show that the electric field depends on the distance from the charge, but feel free to just say E).

So E • dA = E(r) * 4 * pi * r^2

E(r) * 4 * pi * r^2 = Qenc/

E(r) = Qenc / 4 * pi * r^2 *

You can see that the electric field due to a point charge depends on 1/r^2

Hope that is helpful!