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This is a question from Irodov:

Two small equally charged spheres, each of mass $m$ are suspended from the same point silk threads length $l$. The distance between the spheres $x\ll l$. Find the rate $\mathrm dq/\mathrm dt$ with which the charge leaks each sphere if their approach velocity varies as $v = a/\sqrt x$, where $a$ is a constant.

In the solution, the professor considered steady state condition for one of the spheres, after drawing the force diagram for one of the spheres, he used the horizontal and vertical equilibrium condition to find $q$ in terms of $x$ and differentiated it to get the rate of leak of charge.

My question is that if we know that the mass is constantly changing velocity and hence accelerating, how can we say it's in equilibrium state?

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  • $\begingroup$ Hello! I have edited your question using MathJax (LaTeX) math typesetting. For future questions, you can refer to MathJax basic tutorial and quick reference. Thanks! $\endgroup$
    – jng224
    Commented Jul 16, 2021 at 8:56
  • $\begingroup$ Thank you. This was my first question here so i didn't know how exactly i can do it $\endgroup$
    – Varun Upreti
    Commented Jul 17, 2021 at 9:27

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The question is ambiguous, but here is my interpretation:

How is it equilibrium if velocity is a function of x? Actually, since on a microscopic scale charge is quantised it shoudn't be dq/dt but rather $\frac{\Delta q}{\Delta t}$. Now what Irodov is asking is that: Find $\Delta x$ when $\Delta q$ charge leaks out in $\Delta t$ seconds. Here $\Delta x$ is crucially the distance between equilibrium states. Essentially he is asking how much will the equilibrium distance change when charge is leaked. This should have been mentioned in the question, but I hope my answer clears your ambiguity.

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