Two identical spheres with mass m are hung from silk threads of length L, as sho

Question

Two identical spheres with mass m are hung from silk threads of length L, as shown in the figure (Figure 1) . Each sphere has the same charge, so q1=q2=q. The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges.

Suppose that the angle is small, and find the equilibrium separation d between the spheres (Hint: If is small, then tansin.)

Express your answer in terms of the variables q, L, m and appropriate constants.

Explanation / Answer

mass of balls, = m
lenghth of threads = L
charge on each sphere q1 = q2 = q
angle theta is small , and so is the radius of each sphere
so, electrostatic force of repuslion, F = kq^2/d^2
where sin(theta) = d/2L
so, d = 2Lsin(theta)
hence F = kq^2/4L^2sin^2(theta)

so let tension in the string be T
then from force balance
Tcos(theta) = mg
Tsin(tjheta) = kq^2/d^2

tan(thetA) = KQ^2/mgd^2
d = sqroot(kq^2/mg*tan(theta))
for small angles
tan (thea) = sin(theta)
but sin(theta) = d/2L
so, d = sqroot(kq^2/mg*sin(theta)) = sqroot(kq^2*2L/mg*d)
d^2 = kq^2*2L/mgd
d^3 = kq^2*2L/mg
d = cuberoot(kq^2*2L/mg)
here k is coloumbs constant, g is acceleration due to gravity

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