Solving Dissipating Charge Physics Problem

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The discussion revolves around solving a physics problem involving two charged spheres suspended by threads. The user initially struggles with the equations of motion and forces acting on the spheres, particularly in relating charge leakage to their approach velocity. After attempting various methods, including differentiating the force equation and considering angular speed, they find the problem complex and unsolvable in its current form. Guidance is provided to simplify the approach by approximating the angle and differentiating the charge with respect to time. Ultimately, the correct solution for the rate of charge leakage is presented as dq/dt = (3/2)a√(2πε₀mg/ℓ).
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A friend of mine has given me a physics problem to solve. Here it is:
Two small equally charged spheres, each of mass m, are suspended from the same point by silk threads of length \ell. The distance between the spheres x << \ell. 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.​
I started by writing down the forces:
mg\tan{\theta} = \frac{q^2}{4\pi \epsilon_0 x^2}
Since x << \ell, I concluded that \sin{\theta} \approx \tan{\theta}. I rewrote the above force equation with sin instead of tan.

After this, I started doing things randomly. At first, I tried using angular speed with \ell as the radius of rotation, but that seemed unnecessarily complicated.

So I went back and differentiated both sides of the first equation. I don't know if this is correct, but I let \cos{\theta} be 1 since \theta \approx 0. So, using the quotient rule, I had:
mg = \frac{1}{2\pi \epsilon_0}\frac{q^2(\frac{dx}{dt}) - q\frac{dq}{dt}}{x^3}
At this point, since \frac{dx}{dt} = \frac{a}{\sqrt{x}}, I started substituting things in. However, I ended up with charge as a function of time, distance as a function of time, and the time derivative of the charge function all in one equation - unsolvable.

The other way that I tried it was to start from the premise that \ell \frac{d\theta}{dt} = \frac{a}{\sqrt{x}} and go from there. But I don't know if this is the correct way either.

Basically, I have no idea what I'm doing. My friend said this is an easy problem, but I am stumped. Am I approaching this incorrectly?

Thanks for the assistance.
 
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Solve your original equation for q and approximate \theta with x/l. Then differentiate q with respect to time. You should end up with dq/dt being constant within this approximation.

[Note: you may need to make minor adjustments to the preceding depending on whether your angle is the half-angle or full angle.]
 
I don't think one should assume that the spheres are in equilibrium. That is

T\sin(\theta) - F_Q = m\ddot{x}

since

v_x(x) \Rightarrow v_x(t)
 
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Tide said:
Solve your original equation for q and approximate \theta with x/l. Then differentiate q with respect to time. You should end up with dq/dt being constant within this approximation.

[Note: you may need to make minor adjustments to the preceding depending on whether your angle is the half-angle or full angle.]
Thanks - I have found my errors now. I had to approximate \theta with \frac{x}{2\ell}.

For those who are interested, the answer is:
<br /> \frac{dq}{dt} = \frac{3}{2}a\sqrt{\frac{2\pi \epsilon_0 mg}{\ell}}<br />
 
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