Problem — Collision between charges with friction

Problem

Two particles ($q_1 = 10^{-3}\ \text{C}$, $m_1 = 1\ \text{kg}$; $q_2 = -10^{-4}\ \text{C}$, $m_2 = 2\ \text{kg}$) are initially at a distance $d = 10\ \text{m}$. A friction force $F_a = 5\ \text{N}$ acts between them, dissipative and constant along the path. The charges attract each other: do they meet? Use an energy table with $E_{\text{cin}}$, $E_{\text{pot,el}} = k\,q_1 q_2/r$ and $E_{\text{term}}$; compare the energy released by the approach with the work done by friction.

Solution

Using $k = 8{,}99\cdot 10^9\ \text{N·m}^2/\text{C}^2$. The charges have opposite signs, so they attract and the electrostatic potential energy is negative:

$E_{\text{pot,el}}(r) = k\,\frac{q_1 q_2}{r},\qquad q_1 q_2 = (10^{-3})(-10^{-4}) = -10^{-7}\ \text{C}^2$

Balance (bubble diagram): the approach releases potential energy (which becomes kinetic energy), while friction dissipates some of it as heat. The charges meet if, as they approach, the potential energy released exceeds the work done by friction.

Potential energy at $r = 10\ \text{m}$:

$E_{\text{pot,el}}(10) = 8{,}99\cdot 10^9 \cdot \frac{-10^{-7}}{10} = -89{,}9\ \text{J}$

As they approach, $r$ decreases and $E_{\text{pot,el}} = k q_1 q_2/r \to -\infty$: the energy released is unbounded. Quantitative comparison closing from $10\ \text{m}$ down to, say, $1\ \text{m}$:

$\Delta E_{\text{lib}} = |E_{\text{pot,el}}(1) - E_{\text{pot,el}}(10)| = k\,|q_1 q_2|\left(\frac{1}{1} - \frac{1}{10}\right) = 899\cdot 0{,}9 \approx 809\ \text{J}$

Work done by friction over the initial $10\ \text{m}$:

$W_a = F_a\, d = 5\cdot 10 = 50\ \text{J}$

Already, closing by just $9\ \text{m}$ releases $\sim 809\ \text{J} \gg 50\ \text{J}$; as they get closer the available energy grows without bound while friction dissipates at most $\sim 50\ \text{J}$ over the whole path. The attraction wins clearly.

$\ev{\text{Yes, they meet: Coulomb energy released } (\gtrsim 800\ \text{J}) \gg W_a = 50\ \text{J}}$

Links

Topics: Electrostatics Concepts: Electrostatic potential energy · Kinetic energy Skills: Conservation of energy Methods: Energy-exchange bubble diagram