Question

Can I calculate surface areas and volumes of weird shapes? :O

Answer 1

Let's say you have 3 charges and the electric flux is exactly the same value at every point on this surface enclosing them:

So I say now:

\[\iint_S \vec E \cdot \vec N dS = \frac{Q}{\varepsilon_0}\]
Since \( \vec E \cdot \vec N\) is just some constant, I pull it out, and all I'm left with is the surface area:

\[A=\iint_S dS = \frac{Q}{\varepsilon_0 (\vec E \cdot \vec N)}\]

So now I imagine that this quantity here which was a constant actually represents a family of different possible surface areas with this constant that are each held inside the other, so I rename this:

\[\vec E \cdot \vec N = t\]

And now I have parametrized some surface areas:

\[A(t) = \frac{Q}{\varepsilon_0 t}\]

and I can now integrate over a chunk of this to get the volume of some of this weird shape:

\[V = \int_a^b \frac{Q}{\varepsilon_0 t} dt = \frac{Q}{\varepsilon_0 } \ln \frac{b}{a}\]

I dunno can we get more info out of this?