To determine the distance \( l \) between the two parallel reflecting glass sheets Y and Z, we need to consider the phenomenon of interference, which occurs when light waves overlap and combine. In this scenario, white light is incident on sheet Y, and the light that reflects off both sheets will create an interference pattern. The appearance of a dark band in the spectrometer indicates that destructive interference is occurring at a specific wavelength.
Understanding Interference in Thin Films
When light reflects off the surfaces of the glass sheets, some of it reflects from the top surface (Y) and some from the bottom surface (Z). The light waves that reflect from these surfaces can interfere with each other. For destructive interference to occur, the path difference between the two waves must be an odd multiple of half the wavelength of the light.
Path Difference Calculation
The path difference \( \Delta \) between the two waves can be expressed as:
- When light reflects off a medium with a higher refractive index (like glass), it undergoes a phase change of \( \pi \) (or half a wavelength).
- The total distance traveled by the light reflecting off Z is \( 2l \) (it travels down to Z and back up to Y).
Thus, the effective path difference considering the phase change is:
Path Difference, \( \Delta = 2l + \frac{\lambda}{2} \)
Condition for Destructive Interference
For destructive interference to occur, the path difference must satisfy the condition:
Condition: \( \Delta = (m + \frac{1}{2})\lambda \)
Here, \( m \) is an integer (0, 1, 2, ...), and \( \lambda \) is the wavelength of the light. Setting the two expressions for path difference equal gives:
2l + \frac{\lambda}{2} = (m + \frac{1}{2})\lambda
Solving for Distance \( l \)
Rearranging this equation allows us to isolate \( l \):
2l = (m + \frac{1}{2})\lambda - \frac{\lambda}{2}
Which simplifies to:
2l = m\lambda
Thus, we find:
l = \frac{m\lambda}{2}
Final Thoughts
In summary, the distance \( l \) between the two glass sheets must be equal to \( \frac{m\lambda}{2} \) for the observed dark band at the wavelength \( \lambda \). The specific value of \( m \) will depend on which order of interference is being observed. For the first dark band, \( m \) would be 1, leading to \( l = \frac{\lambda}{2} \). This relationship illustrates how the properties of light and the geometry of the setup interact to produce observable phenomena in optics.
