# 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 [ ].

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(c) Doug Davis, 2002; all rights reserved