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Can spheres leaking charge be assumed to be in equilibrium?

Physics Asked by user27657 on December 16, 2020

I am struggling with the following problem (Irodov 3.3):

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

This is embarrassingly simple; we make an approximation for $x ll l$ and get
$$
frac{1}{4 pi epsilon_0} frac{q^2}{x^2} – frac{mgx}{2l} = m ddot{x}.
$$
We can get $ddot{x}$ from our relation for $v$, so we can solve for $q$ and then find $frac{dq}{dt}$.

However, in general, $frac{dq}{dt}$ will depend on $x$ and hence on $t$. The answer in the back of the book and other solutions around the web have $frac{dq}{dt}$ a constant.

You can get this by assuming that at each moment the spheres are in equilibrium, so that you have $ddot{x} = 0$ in the equation of motion above.

Does the problem tacitly imply we should assume equilibrium and hence $frac{dq}{dt}$ is constant, or am I missing something entirely? I.e. why is the assumption of equilibrium justified? I understand reasoning like “the process happens very gradually, so the acceleration is small compared to other quantities in the problem,” but I don’t understand how that is justified by the problem itself, where we are simply given that the spheres are small (so we can represent them as points) and $x ll l$ (which we have used to approximate the gravity term in the equation of motion).

2 Answers

I think the answer is sonething that you have overlooked, a (. ) AKA FULL STOP.

You state that the web results say the answe is dt/dq a this is a constant because (a) is A constant.

The question you ask is " Does the problem tacitly imply we should assume equilibrium and hence dt/dq is constant, or am I missing something entirely?"

I reckon you've overlooked the fact its (a) NOT (dt/dq) thats the constant.

Have I answered your question?

Sorry for the technical lomotations of my keyboard.

Answered by Billiehuman on December 16, 2020

If we continue with the suggestion you made, of obtaining $ddot x$ from the equation of motion $v=a/sqrt{x}$ which was provided, and substituting this into the equation $F=mddot x$, then we do indeed find that $dot q$ is not constant. It is only by ignoring the $mddot x$ term - by assuming that $vapprox 0$ - that we can reach the result which Irodov intended.

But there is nothing in the question statement which justifies the assumption that $v approx 0$. No values are given which would enable us to conclude that $dot v=-(a/2xsqrt{x})v$ can be neglected so that there is a quasi-static equilibrium.

The conclusion must be that Irodov made an error. He deliberately imposed an unrealistic but fairly simple equation of motion $v=a/sqrt{x}$ in order to derive an equally unrealistic but simple result (that $dot q$ is constant). While doing so he failed to state the assumptions which were necessary to obtain this result.

Even the most respected authors and textbooks are fallible.

Answered by sammy gerbil on December 16, 2020

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