A source S of monochromatic light and a detector D are both...

A source $S$ of monochromatic 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

Destructive Interference

Destructive interference happens when the phase difference between two waves is an odd multiple of ?, which causes the amplitudes to cancel each other out. In practical terms, this is achieved when the path difference (taking into account any phase changes from reflection) corresponds exactly to an integer number of wavelengths, leading to dark fringes.

Constructive Interference

Constructive interference occurs when the phase difference between two waves is an integer multiple of 2?, or equivalently when the effective path difference is an odd multiple of half-wavelengths that accommodates any phase change on reflection. This results in reinforcement of the wave amplitudes, producing a brighter interference pattern.

Wave Interference

Wave interference occurs when two or more waves superimpose, leading to a new wave pattern. In optics, interference is a key concept to understand phenomena such as the formation of fringes or the variation in light intensity, as waves can constructively add to produce brighter regions or destructively cancel to produce dark regions.

Path Difference

Path difference is the extra distance one wave travels relative to another before they meet at a point. It plays a critical role in interference because the phase difference, which determines whether the interference is constructive or destructive, is directly related to the path difference between the interfering beams.

Phase Change on Reflection

When a light wave reflects off a medium with a higher refractive index, it undergoes a phase change of ? (equivalent to a half-wavelength shift). This phase shift must be considered when calculating the effective path difference between reflected and direct rays, as it fundamentally alters the conditions for constructive and destructive interference.

Transcript

Please give Ace some feedback