Problem — Challenge: equilibrium of three collinear charges

Problem

Challenge: equilibrium of three collinear charges. Three charges in a line: $+q$ at one point, $-Q$ at another distance, and $+q'$ placed on the connecting segment at distance $r_1$ from $+q$ and $r_2$ from $-Q$. Find the condition on $r_1$ and $r_2$ for $+q'$ to be in equilibrium. Briefly discuss stability.

Solution

Forces on the charge $+q'$. $q'$ experiences repulsion from $+q$ and attraction from $-Q$. The equilibrium condition requires the two magnitudes to be equal:

$k\,\frac{q\,q'}{r_1^2} = k\,\frac{Q\,q'}{r_2^2}$

Simplification. Cancelling $k$ and $q'$:

$\frac{q}{r_1^2} = \frac{Q}{r_2^2} \quad\Rightarrow\quad \frac{r_2^2}{r_1^2} = \frac{Q}{q} \quad\Rightarrow\quad \frac{r_2}{r_1} = \sqrt{\frac{Q}{q}}$

Stability. Even when equilibrium along the line is imposed, for a small transverse perturbation (perpendicular to the axis) the charge does not return: the equilibrium is unstable. This is a manifestation of Earnshaw’s theorem, according to which no configuration of point charges under purely electrostatic forces can be in stable equilibrium.

$\ev{\dfrac{r_2}{r_1} = \sqrt{\dfrac{Q}{q}}\ \text{(equilibrio instabile)}}$

Linked atoms

Topics: Elettrostatica Concepts: Legge di Coulomb Skills: Principio di sovrapposizione · Impostazione simbolica