Gauss' Law
Problem Solution
24.** A slab of insulating material (infinite in two of its three dimensions) has a uniform positive charge density rho . An edge view of the slab is shown here.
(a) Show that the electric field a distance x from its center and inside the slab is E = x / .
(b) Suppose an electron of charge - e and mass m is placed inside the slab. If it is released from rest at a distance x from the center, show that the electron exhibits simple harmonic motion with a frequency
For the mass-and-spring simple harmonic oscillator, we had
F = - k x and found the frequency to be
f = [ 1 / 2 ] SQRT [ k / m ] We have found the electric field to be
E = x / So the electric force on an electron is
F = - [ x / ] e = - ( e / ) x That means we have a simple harmonic oscillator with
k = ( e / ) so its frequency is
f = [ 1 / 2 ] SQRT [ k / m ] f = [ 1 / 2 ] SQRT [ ( e / ) / m ]
f = [ 1 / 2 ] SQRT[ e / m ]
f = [ 1 / 2 ] SQRT[ e / m ]
Remember, SQRT[ ] means the SQUARE ROOT of [ ].
(c) Doug Davis, 2002; all rights reserved
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