How do I find the distance #r# separating the spheres given the following...?

#*# Length of the string(#L#) #=1.5m#

#*# The charge on each of the spheres is #10# nanoCoulombs.

#*# Mass of both spheres #(m)=43# grams

Also, assume that the angle (#theta#) between the strings is very small. (#sintheta~~tantheta#)

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1 Answer
Ananda Dasgupta
Mar 7, 2018

1.5 cm, roughly

Explanation:

Distance between the spheres #D = 2L sin(theta/2) ~~ L theta# , so #theta ~D/L#

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So, the Coulomb force of repulsion

#F = 1/{4 pi epsilon_0} {q_1q_2}/D^2#

Since the spheres are in equilibrium, the resultant of #vec F# and #m vec g# must cancel out the tension in the string, which has to be along the string. Thus

#F/{mg } = tan theta ~~ theta ~~ D/L#

so that

#1/{4 pi epsilon_0} {q_1q_2}/D^2 = {mgD}/L#

So, we have

#D^3 = 1/{4 pi epsilon_0} {q_1q_2}/{mg} L= (9 times 10^9) times (10^-8)^2/{0.043times 9.81} times 1.5 \mbox{m}^3 ~~3.2 times 10^-6 \mbox{m}^3 #

Thus #D~~0.015 # m # = 1.5# cm

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