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The solution to problem 20.54 in the cphy 122 course taught by h. L. Neal. The problem deals with the conservation of electric charge and the calculation of charges on two identical spherical conductors when they are brought together. The fundamental principle of charge conservation, derives the equation for the charge on each sphere, and provides the solutions for the charges. Useful for students studying electrostatics and electric charge.
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The conservation of electric charge is essential to solving this problem. One statement
of this fundamental principle is that the total charge of a closed system is always constant.
Here the system is represented by the two identical spherical conductors. Initially we have
the total charge as q 1 + q 2. When the two spheres are brought together, the total charge is
rearranged, so that each sphere has the same charge q. This gives the รnal total charge as
2 q. Conservation of charge requires the q 1 + q 2 = 2q. Since the magnitude of the repulsive
force is
F = k
q^2
r^2
the charge q is
q = r
p F=k:
For the attractive force of magnitude F we must have q 1 q 2 < 0 , so that
q 1 q 2 = q 2 :
This last equation may be transformed into the quadratic equation for q 1 (or q 2 ) by using
the conservation of charge equation to write q 2 in terms of q 1 :
q 1 (2q q 1 ) = q 2 :
Moving everything to the right-hand side of the equation gives
q 2 1 ^2 qq^1 ^ q
2 = 0:
The solutions for q 1 are
q 1 =
2 q
p 4 q^2 + 4q^2
= q
p 2
It should be easy for you to show that
q 1 = q
p 2
q 2 = q
p 2