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Unformatted text preview: 19. The electric ﬁeld at a point on the axis of a uniformly charged ring, a distance z from the ring center,
is given by
4πε0 (z 2 + R2 )3/2
where q is the charge on the ring and R is the radius of the ring (see Eq. 23–16). For q positive, the ﬁeld
points upward at points above the ring and downward at points below the ring. We take the positive
direction to be upward. Then, the force acting on an electron on the axis is
For small amplitude oscillations z eqz
4πε0 (z 2 + R2 )3/2 R and z can be neglected in the denominator. Thus,
F =− eqz
4πε0 R3 The force is a restoring force: it pulls the electron toward the equilibrium point z = 0. Furthermore, the
magnitude of the force is proportional to z , just as if the electron were attached to a spring with spring
constant k = eq/4πε0 R3 . The electron moves in simple harmonic motion with an angular frequency
where m is the mass of the electron. ...
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