16—Calculus of Variations 391At this point pick a form for the index of refraction that will make the integral easy and will still plausibly
represent reality. The index increases gradually above the road surface, and the simplest function works:
f(y) =n 0 (1 +αy). The index increases linearly above the surface.
x(y) =
∫
C
√
n^20 (1 +αy)^2 −C^2
dy=
C
αn 0
∫
dy
1
√
(y+ 1/α)^2 −C^2 /α^2 n^20
This is an elementary integral. Letu=y+ 1/α, thenu= (C/αn 0 ) coshθ.
x=
C
αn 0
∫
dθ ⇒ θ=
αn 0
C
(x−x 0 ) ⇒ y=−
1
α
+
C
αn 0
cosh(
(αn 0 /C)(x−x 0 )
)
Candx 0 are arbitrary constants, andx 0 is obviously the center of symmetry of the path. You can
relate the other constant to they-coordinate at that same point:C=n 0 (αy 0 + 1).
Because the value ofαis small for any real roadway, look at the series expansion of this hyperbolic
function to the lowest order inα.
y≈y 0 +α(x−x 0 )^2 / 2 (16.25)
When you look down at the road you can be looking at an image of the sky. The light comes from
the sky near the horizon down toward the road at an angle of only a degree or two. It then curves up
so that it can enter your eye as you look along the road. The shimmering surface is a reflection of the
distant sky or in this case an automobile — a mirage.
*
16.5 Electric Fields
The energy density in an electric field is 0 E^2 / 2. For the static case, this electric field is the gradient
of a potential,E~=−∇φ. Its total energy in a volume is then
W=
0
2
∫
dV(∇φ)^2 (16.26)
What is the minimum value of this energy? Zero of course, ifφis a constant. That question is too
loosely stated to be much use, but keep with it for a while and it will be possible to turn it into
something more precise and more useful. As with any other derivative taken to find a minimum, change
the independent variable by a small amount. This time the variable is the functionφ, so really this
quantityWcan more fully be written as a functionalW[φ] to indicate its dependence on the potential
function.
W[φ+δφ]−W[φ]=
0
2
∫
dV
(
∇(φ+δφ)
) 2
−
0
2
∫
dV(∇φ)^2
=
0
2
∫
dV
(
2 ∇φ.∇δφ+ (∇δφ)^2
)
* Donald Collins, Warren Wilson College