What is the space between two consecutive grooves on the CD

[thuanthuan]

Stand a couple of meters from a lightbulb.
Facing away from the light,
hold a compact disc about 10 cm
from your eye and tilt it until the reflection
of the bulb is located in the
hole at the disc’s center. You should
see spectra radiating out from the
center, with violet on the inside and
red on the outside. Now move the
disc away from your eye until the violet
band is at the outer edge. Carefully
measure the distance from your
eye to the center of the disc and also
determine the radius of the disc. Use
this information to find the angle a to
the first-order maximum for violet
light. Now use Equation 38.10 to determine
the spacing between the
grooves on the disc. The industry
standard is 1.6
m. How close did
you come?

( Fundamentals of Physics, p1226, 7th edition)

Can you tell me how to perform this experiment ? .. and also prove that it's true.
I don't understand it.
Thank you.

 

[thuanthuan]

note that equation 3.10: for diffraction grating, the condition for maxima in the interference pattern at the angle a is d.sina=m.lambda
(d = slit spacing. m=0,1,2... - order number, lambda=wavelength)

 

[charng]

To perform this experiment, follow these steps:

1. Stand a couple of meters away from a lightbulb, facing away from it.
2. Hold a compact disc about 10 cm from your eye and tilt it until the reflection of the bulb is located in the hole at the disc's center.
3. Observe the spectra radiating out from the center, with violet on the inside and red on the outside.
4. Move the disc away from your eye until the violet band is at the outer edge.
5. Carefully measure the distance from your eye to the center of the disc and also determine the radius of the disc.
6. Use this information to find the angle a to the first-order maximum for violet light.
7. Use Equation 38.10 (which relates the angle of diffraction to the wavelength of light and the spacing between the diffraction grating) to determine the spacing between the grooves on the disc.
8. Compare your result to the industry standard of 1.6 μm.

To prove that this is true, we can use the equation given in the textbook, which is based on the principles of diffraction. Diffraction is the bending of light as it passes through a narrow slit or around an obstacle. In this case, the grooves on the CD act as a diffraction grating, causing the light to diffract and produce the spectra we observe.

The equation, known as the grating equation, is given by:

a sinθ = mλ

Where:
a is the spacing between the grooves
θ is the angle of diffraction
m is the order of the diffraction maximum
λ is the wavelength of light

In our experiment, we are measuring the distance from our eye to the center of the disc (r) and the radius of the disc (R). From this, we can calculate the angle of diffraction using the following formula:

sinθ = r/R

Substituting this into the grating equation, we get:

a = mλR/r

Using the values we measured in the experiment, we can calculate the spacing between the grooves on the disc. Comparing this to the industry standard of 1.6 μm, we can see how close our result is to the expected value.

In conclusion, this experiment is based on the principles of diffraction and the grating equation, and by performing it and using the appropriate calculations, we can determine the spacing