Problem 22 - University Physics with Modern Physics

Question

Two slits spaced 0.260 \(\mathrm{mm}\) apart are placed 0.700 \(\mathrm{m}\) from a screen and illuminated by coherent light with a wavelength of 660 nm. The intensity at the center of the central maximum \(\left(\theta=0^{\circ}\right)\) is \(I_{0}\) . (a) What is the distance on the screen from the center of the central maximum to the first minimum? (b) What is the distance on the screen from the center of the central maximum to the point where the intensity has fallen to \(I_{0} / 2 ?\)

Step-by-Step Solution

Answer

(a) 0.89 mm; (b) 0.445 mm.

Step 1: Understand the Problem

This problem involves a double slit experiment, where we need to find the distances on a screen where certain intensity conditions are met. We have the slit separation, distance to the screen, and wavelength of the light given.

Step 2: Calculate the First Minimum in Double Slit Interference

For the first minimum, we use the condition for destructive interference, which is given by \( d \sin \theta = \left( m + \frac{1}{2} \right) \lambda \) for the first minimum \( m=0 \). Substituting \( d = 0.260 \times 10^{-3} \) m and \( \lambda = 660 \times 10^{-9} \) m yields: \[ 0.260 \times 10^{-3} \sin \theta = 0.5 \times 660 \times 10^{-9} \]Solving for \( \theta \):\[ \sin \theta = \frac{0.5 \times 660 \times 10^{-9}}{0.260 \times 10^{-3}} \approx 0.00127 \]Thus, \( \theta \approx 0.0728^{\circ} \).

Step 3: Convert the Angle to Distance on the Screen

Using the small angle approximation, \( \sin \theta \approx \tan \theta \approx \theta \), we can calculate the distance \( x \) on the screen as:\[ x = L \cdot \theta = 0.700 \cdot 0.0728 \times \frac{\pi}{180} \approx 0.00089 \text{ m} = 0.89 \text{ mm} \]

Step 4: Determine Intensity Half-Maximum Condition

The intensity in a double slit experiment is given by \( I = I_0 \cos^2\left(\frac{\pi d \sin \theta}{\lambda}\right) \). For the intensity to be \( I_0/2 \), we need:\[ \cos^2\left(\frac{\pi d \sin \theta}{\lambda}\right) = \frac{1}{2} \]This gives:\[ \frac{\pi d \sin \theta}{\lambda} = \frac{\pi}{4} \text{ or } \frac{3\pi}{4} \]

Step 5: Solve for the Half-Maximum Angle

We use \( \frac{\pi d \sin \theta}{\lambda} = \frac{\pi}{4} \):\[ d \sin \theta = \frac{\lambda}{4} \]Plugging values, \( d = 0.260 \times 10^{-3} \) m and \( \lambda = 660 \times 10^{-9} \) m:\[ 0.260 \times 10^{-3} \sin \theta = \frac{660 \times 10^{-9}}{4} \]\[ \sin \theta = \frac{165 \times 10^{-9}}{0.260 \times 10^{-3}} \approx 0.000635 \]Thus, \( \theta \approx 0.0364^{\circ} \).

Step 6: Convert the Half-Maximum Angle to Distance on the Screen

Calculate the distance \( x \) using the angle found:\[ x = L \cdot \theta = 0.700 \cdot 0.0364 \times \frac{\pi}{180} \approx 0.000445 \text{ m} = 0.445 \text{ mm} \]

Key Concepts

Interference Patterns

In the double slit experiment, interference patterns are a fascinating phenomenon created when two or more waves overlap and combine. This results in alternating regions of high and low intensity on a screen. These patterns give us a visual representation of how waves interact. In physics, it showcases the wave nature of light.

The interference effects are due to the constructive and destructive interference of light waves. Constructive interference occurs when the crest of one wave aligns with the crest of another, intensifying the light. Consequently, the screen shows bright spots called maxima. On the other hand, destructive interference happens when the crest of one wave meets the trough of another, which cancels the light out, producing dark areas known as minima.

Thus, every double slit experiment displays a series of bright and dark bands, known as interference fringes. These patterns depend on factors like slit separation, wavelength of the light used, and the distance between the slits and the screen.

Wavelength of Light

The wavelength of light is a crucial parameter in the double slit experiment. It is the distance over which the wave's shape repeats, determining the color of the light when visible. In our exercise, the light has a wavelength of 660 nm (nanometers), which falls in the red region of the visible spectrum.

This wavelength significantly influences the spacing of the interference fringes. Generally, a longer wavelength results in fringes that are spaced further apart, while shorter wavelengths produce more closely spaced fringes. In the double slit experiment, the relationship between the wavelength and the interference pattern can be described mathematically by the interference condition:

• For maxima, the condition is: \(d \sin \theta = m \lambda\)
• For minima, it's given by: \(d \sin \theta = (m + \frac{1}{2}) \lambda\)

where \(d\) is the slit separation, \(\theta\) is the angle of the interference fringe, and \(m\) is the order of the fringe.

Hence, understanding wavelength allows us to predict where different fringes will appear on the screen, making it essential for performing and interpreting interference experiments.

Destructive Interference

Destructive interference is a key concept that explains how waves can cancel each other out. In a double slit experiment, it occurs when two light waves are out of phase by half a wavelength (or an odd multiple of half-wavelengths). This phase difference leads to the waves cancelling each other's energy, resulting in a decrease or complete nullification of light's intensity at certain points.

These points manifest as dark lines or bands on the interference pattern known as minima. The condition for destructive interference can be expressed as:

• \(d \sin \theta = (m + \frac{1}{2}) \lambda\)

where \(d\) is the separation between slits, \(\lambda\) is the wavelength of the light, and \(\theta\) is the angle relative to the central maximum. The \(m\) is an integer that signifies the order of the minimum.

Destructive interference explains why certain areas on the screen are dark, as the negative overlap of the wave frequencies imperceptibly reduces the light intensity there. Understanding these concepts allows scientists and students to grasp the fundamental principles of wave behavior in physics.