41. The particle with charge −qhas both potential and kinetic energy, and both of these change when the
radius of the orbit is changed. We first find an expression for the total energy in terms of the orbit radius
r.Qprovides the centripetal force required for −qto move in uniform circular motion. The magnitude
of the force is F=Qq/4πε0r2. The acceleration of −qis v2/r,wherevis its speed. Newton’s second
law yields
Qq
4πε0r2=mv2
r=⇒mv2=Qq
4πε0r,
and the kinetic energy is K=1
2mv2=Qq/8πε0r. The potential energy is U=−Qq/4πε0r,andthe
total energy is
E=K+U=Qq
8πε0r−Qq
4πε0r=−Qq
8πε0r.
When the orbit radius is r1the energy is E1=−Qq/8πε0r1andwhenitisr2the energy is E2=
−Qq/8πε0r2. The difference E2−E1is the work Wdone by an external agent to change the radius:
W=E2−E1=−Qq
8πε01
r2
−1
r1=Qq
8πε01
r1
−1
r2.
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