Problem 55 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

The longest wavelength for constructive interference is 72 cm.

Step 1: Understanding Path Difference

For constructive or destructive interference, the key is to understand the path difference between the two light rays: one traveling directly from the source to the detector, and the other reflecting off the glass. The path taken by the reflected ray can be expressed as the hypotenuse of a right triangle where the horizontal distance is \(x\) and the vertical distance is \(2h\). Hence, the path length is \(\sqrt{x^2 + (2h)^2}\).

Step 2: Calculate Path Difference

The path difference between the reflected ray and direct ray is \(\Delta = \sqrt{x^2 + (2h)^2} - x\). This value accounts for the extra distance traveled by the reflected wave compared to the direct wave.

Step 3: Phase Change Consideration

When light reflects off a surface like glass, it undergoes a phase shift of \(\pi\) rad (or equivalently half a wavelength). This phase change must be accounted for in the interference condition.

Step 4: Constructive Interference Condition

Constructive interference occurs when the path difference is an odd multiple of half-wavelengths: \(\Delta = (m + 0.5)\lambda\), where \(m\) is an integer. Substituting the expression for \(\Delta\), we get \(\sqrt{x^2 + 4h^2} - x = \left(m + \frac{1}{2}\right)\lambda\).

Step 5: Destructive Interference Condition

Destructive interference occurs when the path difference is an integer multiple of wavelengths: \(\Delta = m\lambda\). Hence, \(\sqrt{x^2 + 4h^2} - x = m\lambda\).

Step 6: Solving for Longest Wavelength

To find the longest wavelength for constructive interference with given \(h = 24 \text{ cm}\) and \(x = 14 \text{ cm}\), we use the constructive interference condition: \(\sqrt{14^2 + 4 \times 24^2} - 14 = \left(m + \frac{1}{2}\right) \lambda\). Solving \(\sqrt{196 + 2304} = 50\), the equation becomes \(50 - 14 = \left(m + \frac{1}{2}\right) \lambda\). Therefore, \(36 = \left(m + \frac{1}{2}\right) \lambda\). Selecting \(m = 0\), yields the longest wavelength as \(\lambda = 72 \text{ cm}\).

Key Concepts

constructive interference

In the world of optics, constructive interference is a fascinating phenomenon. It occurs when two or more light waves superimpose in such a way that they amplify each other, resulting in a stronger light intensity. This happens when the crest of one wave aligns perfectly with the crest of another wave. In practical terms, it means the path difference between these waves must be an odd multiple of half-wavelengths.

Specifically, for our context, constructive interference is reached when the path difference \( \Delta \) fulfills the condition \( \Delta = \left(m + \frac{1}{2}\right) \lambda \). Here, \( m \) is an integer, representing the order of the interference, and \( \lambda \) is the wavelength of the light.

By meeting this condition, the light waves combine to produce a brighter, more intense light at the point of interference. This principle is frequently utilized in various applications like antireflective coatings and optical instruments.

destructive interference

Destructive interference is quite the opposite of constructive interference. Instead of amplifying, it results in the cancellation or reduction of the light intensity where two or more light waves overlap. This happens when the crest of one wave aligns with the trough of another, effectively canceling each other out.

For destructive interference, the path difference \( \Delta \) should be an exact multiple of the wavelength: \( \Delta = m \lambda \). Here, \( m \) is again an integer denoting the order of interference.

The net effect is that the amplitude of the resultant wave at the interference point is minimized, often leading to darkness or reduced intensity. This concept is very crucial in fields like noise-canceling technologies and radio wave applications.

phase shift in light waves

When light waves reflect off certain surfaces, like glass, they can undergo a phase shift. This phase shift is essentially a change in the wave's position within its cycle. For light reflecting off a denser medium, this shift is equivalent to \( \pi \) radians, or half a wavelength.

This concept is pivotal in understanding interference because it alters how the waves align when they meet. In the context of our exercise, this phase shift must be factored into interference conditions.

• For the reflected wave, the effective path difference must account for this shift.
• This is why constructive interference occurs at path differences of half-integral wavelengths, rather than integral multiples.

Correctly accounting for phase shifts is crucial for accurate predictions of the interference pattern in practical optics setups.

monochromatic light

Monochromatic light refers to light that consists of a single wavelength or color. Think of it as light with a singular "note" in the spectrum, like the clear tone of a tuning fork.

Using monochromatic light simplifies the study of interference patterns because variations in color (which are due to variations in wavelength) are eliminated. This set wavelength allows for precise calculations and observations.

In laboratory settings or theoretical exercises, monochromatic sources, such as lasers, are often employed to achieve clear and consistent interference patterns without the complexity introduced by multiple wavelengths. Understanding monochromatic light is essential when analyzing and predicting interference, making it a fundamental concept in the study of optics.