Moderate# Graphical Analysis, Calculus and Charge Conservation

EANDM-LQKX4S

Suppose I suspend two charged conducting spheres 10 mm apart along the $x$ axis as shown in the figure below. The radii of the spheres are very small, less than 1 mm.

I place a total charge $Q$ on the spheres, with a charge $q_1$ on sphere one, and a charge $q_2$ on sphere two. The force that sphere two exerts on sphere one is therefore equal to:

$$\vec{F}_{12} = -F_E \hat{x}$$

…where:

$$F_E = \frac{q_1 q_2}{4\pi \epsilon_0 r^2}$$

In this equation, $r$ is the distance between the two charges and $\epsilon_0$ is the vacuum electric permittivity.

I slowly move charge from one sphere to another. The graph below shows the amplitude $F_E$ of this force as a function of the charge $q_1$ on sphere one.

What is the **total charge $Q$** on both spheres?