35.55. A source S of monochrounatic light and a detector D...

35.55. A source $S$ of monochrounatic light and a detector $D$ are both located in air a distance $h$ above a horizontal plane sheet of glass, and are separated by a horizontal distance $x$ . Waves reaching $D$ directly from $S$ interfere with waves that reflect off the glass. The distance $x$ is small compared to $h$ so that the reflection is at close to normal incidence. (a) Show that the condition for constructive interference is $\sqrt{x^{2}+4 h^{2}}-x=\left(m+\frac{1}{2}\right) \lambda,$ and the condition for destructive interference is $\sqrt{x^{2}+4 h^{2}}-x=m \lambda$ (Hint: Take into account the phase change on reflection.) (b) Let $h=24 \mathrm{cm}$ and $x=14 \mathrm{cm} .$ What is the longest wavelength for which there will be constructive interference?

35.55. A source $S$ of monochrounatic light and a detector $D$ are both located in air a distance $h$ above a horizontal plane sheet of glass, and are separated by a horizontal distance $x$ . Waves reaching
$D$ directly from $S$ interfere with waves that reflect off the glass. The distance $x$ is small compared to $h$ so that the reflection is at close to normal incidence. (a) Show that the condition for constructive interference is $\sqrt{x^{2}+4 h^{2}}-x=\left(m+\frac{1}{2}\right) \lambda,$ and the condition for destructive interference is $\sqrt{x^{2}+4 h^{2}}-x=m \lambda$ (Hint: Take into account the phase change on reflection.) (b) Let $h=24 \mathrm{cm}$ and $x=14 \mathrm{cm} .$ What is the longest wavelength for
which there will be constructive interference?

Key Concepts

Geometric Optics

The geometric configuration of the source, detector, and reflecting surface is used to derive the relation between the distances involved and the path difference. By considering the geometry of the light’s trajectory, including the additional distance traveled due to reflection, it is possible to express the interference conditions in terms of the physical parameters (e.g., the horizontal distance and height above the reflecting surface).

Wave Interference

Interference is the phenomenon that occurs when two or more waves meet and superpose, resulting in regions of constructive interference (increased amplitude) and destructive interference (decreased amplitude). This concept is critical in understanding how overlapping light paths from different routes, such as a direct beam and a reflected beam, combine to enhance or diminish overall intensity at a detector.

Path Difference

The path difference between two interfering waves is the extra distance one wave travels relative to the other. In interference problems, this difference determines whether the waves arrive in phase (constructive interference) or out of phase (destructive interference). Calculating the path difference, particularly using geometric relations like the Pythagorean theorem for non-orthogonal propagation, is essential for deriving the interference conditions.

Phase Change Upon Reflection

When light reflects off the boundary between two media with differing refractive indices, it can experience a phase shift, typically of ? (or half a wavelength) when reflecting from a medium of higher refractive index. Accounting for this phase change is vital when determining the overall phase difference between interfering waves, thus affecting the interference condition (whether constructive or destructive).

Transcript

Please give Ace some feedback