The mass required for a slab to buckle is given by
where E is Young's modulus,a is the thickness of the slab,b is the width,l is the length, and g is the gravitational acceleration.
The frequency of oscillation of a slab of mass m is given by
Given an infinite, uniformly charged slab, for outside the slab with
, Gauss's law in an electric field) gives, in cgs,
Use Gauss's law again with one side of the pillbox inside the slab and one out
so
Summarizing,
In MKS, outside the slab with , Gauss's law gives
where
is the permittivity of free space,
so
This is the equation in terms of
and 
angles. To convert to a function of 
,
, and only,use Snell's law
Plugging (18) into (16) yields
Use the trigonometric angle addition formula for sines,
This is the exact formula. For
it also follows that
Taking a binomial expansion with
equal to the term on the left of (21),
which is independent of angle of incidence.
(1)
where E is Young's modulus,a is the thickness of the slab,b is the width,l is the length, and g is the gravitational acceleration.
The frequency of oscillation of a slab of mass m is given by
(2)
Given an infinite, uniformly charged slab, for outside the slab with
, Gauss's law in an electric field) gives, in cgs,(3)
Use Gauss's law again with one side of the pillbox inside the slab and one out
(4)
so
(5)
Summarizing,
(6)
In MKS, outside the slab with
, Gauss's law gives(7)
where
is the permittivity of free space,(8)
so
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
This is the equation in terms of
and angles. To convert to a function of ,, and only,use Snell's law(17)
(18)
Plugging (18) into (16) yields
(19)
Use the trigonometric angle addition formula for sines,
(20)
This is the exact formula. For
(21)
it also follows that
(22)
Taking a binomial expansion with
equal to the term on the left of (21),(23)
(24)
which is independent of angle of incidence.
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