15.3 Solutions
747
or
t × (2R − t) = r
2
∴ t =
r 2
2R
(∵ t << 2R)
(1)
(a) Centres of curvature on the same side.
An air film BC is formed sandwiched between two curved surfaces of
radii R 1 and R 2 in contact at O. The centres of curvature of lenses are
on the same side, Fig. 15.33a. The thickness of air film
t = BC = AC − AB
Fig. 15.33 a Newton’s rings
formed by two curved
surfaces with the centres of
curvature on the same side
Using (1)
AC =
r 2
m
2R 1
;
AB =
r 2
m
2R 2
t =
r 2
m
2
1
R 1
−
1
R 2
(2)
where r m is the radius of the mth ring:
2t = mλ
(dark rings)
(3)
Eliminating t between (2) and (3)
r
2
m
1
R 1
−
1
R 2
= mλ
( m = 0, 1, 2, . . . dark rings) (4)
2t =
m +
1
2
λ
(bright rings)
(5)
r
2
m
1
R 1
−
1
R 2
=
m +
1
2
λ
(m = 0, 1, 2, . . . bright rings)
(6)
(b) Centres of curvature on the opposite side
The surfaces in contact at O are as in Fig. 15.33b; the thickness of air
film t is