CPHY 122
Solution to Problem 20.54
Instructor: H. L. Neal
January 30, 2009
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 q1+q2. When the two spheres are brought together, the total charge is
rearranged, so that each sphere has the same charge q. This gives the โฆnal total charge as
2q. Conservation of charge requires the q1+q2= 2q. Since the magnitude of the repulsive
force is
F=kq2
r2;
the charge qis
q=rpF=k:
For the attractive force of magnitude Fwe must have q1q2<0, so that
q1q2=๎q2:
This last equation may be transformed into the quadratic equation for q1(or q2) by using
the conservation of charge equation to write q2in terms of q1:
q1(2q๎q1) = ๎q2:
Moving everything to the right-hand side of the equation gives
q2
1๎2qq1๎q2= 0:
The solutions for q1are
q1=1
2๎2q๎p4q2+ 4q2๎
=q๎1๎p2๎:
It should be easy for you to show that
q1=q๎1 + p2๎;
q2=q๎1๎p2๎:
1