Problem 9 - Parabolic mirror
(a) Consider a parabolic mirror whose surface satis…es y 2 = 4xf x: A
horizontal light ray that approaches this mirror with a value of y = yo will
strike the mirror at xo = yo2 =4xf : The tangent to the mirror at this location
has a slope given by
q
dy
tan o =
= 2xf =yo = xf =xo :
dx
Now a simple drawing shows that the re‡ected ray has a slope of tan 2 o :
The algebraic expression for this re‡ected light ray is
y
yo = tan 2
where
tan 2
o
2 tan o
=
1 tan2
o
o
(x
xo ) ;
p
p
2 xf xo
2 xf =xo
=
=
:
1 xf =xo
xo xf
The intersection of the x axis by the re‡ected ray occurs when y = 0. Solving
for x yields
p
2 xf xo
p
(x xo )
yo = tan 2 o (x xo ) ! 2 xf xo =
xo xf
xo xf
p
x =
2 xf xo p
+ xo ! x = xo + xf + xo = xf
2 xf xo
This light ray crosses the x axis at x = xf independent of xo and/or yo :
(b) A circle that is tangent to this parabola at x = y = 0 with a radius
R satis…es
(x R)2 + y 2 = R2 :
When x << R we can ignore x2 when compared to R2 with the result:
R2
2xR + y 2 = R2
y 2 = 2xR:
For R = 2xf ; this reduces to
y 2 = 4xf x;
which is the equation for the original parabola.
1
Problem 10 - Lloyd’s Mirror
Lloyd’s mirror is a con…guration (see …gure) which results in interference
between a direct and re‡ected beam from the same source. Assuming a
monochromatic source and D >> d, …nd an expression for the location of
bright fringes on the screen given d and D
Consider a point P on the screen that is a distance y above the mirror.
The distance from the source to the point P is
q
1 (y d)2
2
2
:
r1 = D + (y d) ' D +
2 D
The distance from the image to P is
q
1 (y + d)2
2
2
r2 = D + (y + d) ' D +
:
2 D
The di¤erence in path length is
1
r = r 2 r1 =
(y + d)2 (y d)2
2D
r = 2yd=D
It is useful to note that 2d is the separation of the two light sources. This
since y=D is small, it is approximately equal to sin : This means that within
our approximations, this expression is the same as that for double slit interference. Noting that there is a phase change upon re‡ection at the mirror
the location of the bright fringes satis…es
r = 2yd=D = (m + 1=2)
y
= (m + 1=2) :
D
2d
2