How the electric potential of a charged body depends on the surface area of the body?

Question

I have studied in the book that electric potential of a body depend on the surface area of the body, that is as the surface area increases potential decreases keeping the charge as constant and vice versa, but no explanation for this.

Answer 1

The body you are looking at has a capacitance, where the other capacitor plate is in the theoretical case infinity. If you look at it from a less theoretical aspect it will be the ground or the surrounding objects. It can even be a real capacitor.

Let's look at the equation for capacitance. I'll use the simplest plate configuration, a approach with a spherical object would actually be a better choice for the other plate in infinity case. In this case you simply have two plates where one of them is your object: \begin{align} C=\varepsilon\frac{A}{d} \end{align} And now look at the relation between capacitance and voltage: \begin{align} C=\frac{q}{U} \end{align} You can combine those equations and get: \begin{align} \varepsilon\frac{A}{d}&=\frac{q}{U}\\ U&=\frac{qd}{A\varepsilon} \end{align} So everything other being constant($q$ being constant too) you have the proportion: \begin{align} U\propto \frac{1}{A} \end{align} So for a smaller area you have a higher potential. Imagine for example that you take the insulated capacitor plates and fold them in half. After that you bring them to the same distance.

Answer 2

If the force goes with $q/r^2$ and the surface area with $r^2$ it follows that the force is proportional to the area, since $r^2$ cancels out.

Answer 3

electric potential of a body depend on the surface area of the body, that is as the surface area increases potential decreases 

Exactly. Potential is the "will" for the charges to move away. Like the pressure in a water pipe, potential is what pushes charges to move faster through a conductive path (a wire).

It should be quite simple intuitively that having many equal charges gathered at one small place will make them reject further incoming like charge much more strongly than if the charges were spread over a larger area and weren't as concentrated. The closer you pack like charges the higher the charge concentration is at that point and thus new charges coming close will experience a stronger repelling force to that point.