Worked exercise — Charge on a spring, equilibrium and expansion

Problem

A charge $q_1 = 10^{-4}\ \text{C}$ is fixed at the origin. A second charge $q_2 = 2\cdot 10^{-4}\ \text{C}$ (mass $m = 3\ \text{kg}$) is attached to a spring ($K = 20\ \text{N/m}$, natural length $\ell_0 = 0$) anchored at the origin. 1) Find the equilibrium position. 2) If the system rotates with period $T = 1\ \text{s}$, how does the equilibrium position change?

Solution

Using $k = 9\cdot 10^9\ \text{N·m}^2/\text{C}^2$ (the textbook’s value). The charges have the same sign, so the Coulomb force pushes $q_2$ outward; the spring (with $\ell_0=0$) pulls it towards the centre with force $K x$.

1) Equilibrium (without rotation). Balance between the spring (towards the centre) and Coulomb (outward):

$K x = k\,\frac{q_1 q_2}{x^2} \;\Longrightarrow\; K x^3 = k\,q_1 q_2$

$20\, x^3 = 9\cdot 10^9\cdot 10^{-4}\cdot 2\cdot 10^{-4} = 9\cdot 10^9\cdot 2\cdot 10^{-8} = 180 \;\Longrightarrow\; x^3 = 9$

$\ev{x_{\text{eq}} = \sqrt[3]{9} \approx 2{,}08\ \text{m}}$

2) With rotation (period $T = 1\ \text{s}$). In the rotating frame a centrifugal force $m\omega^2 x$ appears, directed outward, with

$\omega = \frac{2\pi}{T} = 2\pi\ \text{rad/s} \;\Longrightarrow\; m\omega^2 = 3\cdot(2\pi)^2 = 3\cdot 39{,}48 = 118{,}4\ \text{N/m}$

Balance of the inward force (spring) against the outward forces (Coulomb + centrifugal):

$K x = k\,\frac{q_1 q_2}{x^2} + m\omega^2 x \;\Longrightarrow\; (K - m\omega^2)\,x = k\,\frac{q_1 q_2}{x^2}$

The effective stiffness is

$K - m\omega^2 = 20 - 118{,}4 = -98{,}4\ \text{N/m} < 0$

The left-hand side $(K-m\omega^2)x$ is negative (for $x>0$), while the right-hand side $k q_1 q_2/x^2$ is positive: the equation has no positive real solution. Physically the spring is no longer able to hold the charge back: an insufficient spring and the centrifugal force act net outward together with the Coulomb repulsion, so the radius grows without bound.

$\ev{\text{Con } T=1\text{ s: } K - m\omega^2 < 0 \Rightarrow \text{nessun equilibrio, la carica sfugge verso l'esterno}}$

Linked atoms

Topics: Elettrostatica Concepts: Legge di Coulomb Skills: Impostazione simbolica Objects: Molla