Question 24.4: Interference in a Wedge-Shaped Film Goal Calculate interfere...
Interference in a Wedge-Shaped Film
Goal Calculate interference effects when the film has variable thickness.
Problem A pair of glass slides 10.0 \mathrm{~cm} long and with n=1.52 are separated on one end by a hair, forming a triangular wedge of air as illustrated in Figure 24.10. When coherent light from a helium-neon laser with wavelength 633 \mathrm{~nm} is incident on the film from above, 15.0 dark fringes per centimeter are observed. How thick is the hair?
Strategy The interference pattern is created by the thin film of air having variable thickness. The pattern is a series of alternating bright and dark parallel bands. A dark band corresponds to destructive interference, and there is one phase reversal, so 2 n t=m \lambda should be used. We can also use the similar triangles in Figure 24.10 to obtain the relation t / x=D / L. We can find the thickness for any m, and if the position x can also be found, this last equation gives the diameter of the hair, D.
Solve the destructive-interference equation for the thickness of the film, t, with n=1 for air:
t=\frac{m \lambda}{2}
If d is the distance from one dark band to the next, then the x-coordinate of the m th band is a multiple of d :
x=m d
By dimensional analysis, d is just the inverse of the number of bands per centimeter.
d=\left(15.0 \frac{\text { bands }}{\mathrm{cm}}\right)^{-1}=6.67 \times 10^{-2} \frac{\mathrm{cm}}{\text { band }}
Now use similar triangles, and substitute all the information:
\frac{t}{x}=\frac{m \lambda / 2}{m d}=\frac{\lambda}{2 d}=\frac{D}{L}
Solve for D and substitute given values:
D=\frac{\lambda L}{2 d}=\frac{\left(633 \times 10^{-9} \mathrm{~m}\right)(0.100 \mathrm{~m})}{2\left(6.67 \times 10^{-4} \mathrm{~m}\right)}=4.75 \times 10^{-5} \mathrm{~m}
Remarks Some may be concerned about interference caused by light bouncing off the top and bottom of, say, the upper glass slide. It’s unlikely, however, that the thickness of the slide will be half an integer multiple of the wavelength of the helium-neon laser (for some very large value of m ). In addition, in contrast to the air wedge, the thickness of the glass doesn’t vary.