Mathematical Tools for Physics

8—Multivariable Calculus 237

transmitted and reflected at each surface.

b

β

β

β

α

α

α

α

θ

sinβ=nsinα

θ= (β−α) + (π− 2 α) + (β−α)

b=Rsinβ

(30)

The light comes in at the indicated distancebfrom the axis. It is then refracted, reflected, and refracted.
Snell’s law describes the first and third, and the middle one has equal angles of incidence and reflection. The
dashed lines are from the center of the sphere. The three terms in the evaluation ofθcome from the three places
at which the light changes direction, and they are the amount of deflection at each place. The third equation
simply relatesbto the radius of the sphere.
With these three equations, I can try to eliminate the two variablesα andβ to get the single relation
betweenbandθthat I’m looking for. When you do this, you find that the resulting equations are a bit awkward.
It’s sometimes easier to use one of the two intermediate angles as a parameter, and in this case you may choose
to useβ. From the picture you know that it varies from zero toπ/ 2. The third equation givesbin terms ofβ.
The first equation givesαin terms ofβ. The second equation determinesθin terms ofβand theαthat you’ve
just found.
The parametrized relation betweenbandθis then

b=Rsinβ, θ=π+ 2β−4 sin−^1

(

1

n

sinβ

)

, (0< β < π/2) (31)

or you can carry it through and eliminateβ.

θ=π+ 2 sin−^1

(

b

R

)

−4 sin−^1

(

1

n

b

R

)

(32)