Direct Energy, 2018a

266 11.8 Problems
a famous result known as Snell's law.
Consider the gure below. Assume that a ray of light travels from
(x 0 ,y 0 ) to (x 1 ,y 1 ) along the path which takes the shortest time. Material
1 has relative permittivity ɛ r1 , so the light travels in that material
at a constant speed √ c
ɛr1
. Material 2 has relative permittivity ɛ r2 ,so
c
the light travels in that material at a constant speed √
ɛr2
. As we
derived in the Prob. 11.6, the light travels along a straight line in
material 1, and it travels along a straight line in material 2. However,
the lines have dierent slopes as shown in the gure. Assume that
the junction of the two materials occurs at x =0.
(a) Find an equation for the total time it takes the light to travel as
a function of h, the vertical distance at which the path crosses
the y axis. Note that you are nding a function here, F (h),
not a functional. You can use the fact that you know the light
follows a straight line inside each material to nd this function.
(b) The path followed by the light takes the minimum time, so the
derivative dF =0. Use this idea to nd an equation for the
dh
unknown vertical height h. Your answer can be written as a
function of the known constants ɛ r1 , ɛ r2 , x 0 , y 0 , x 1 , y 1 ,and c.
You do not need to solve for h here, but instead just evaluate
the derivative and set it to zero.
(c) Use your result in part b above to derive Snell's law :
√
ɛr1 sin θ 1 = √ ɛ r2 sin θ 2
.
Material 1
ɛ r1
Material2
ɛ r2
(x 1 ,y 1 )
(x 0 ,y 0 )
θ 1
θ 2
(0,h)
x=0