Problem 7 - College Physics

Question

(a) If the wavelength used in a double-slit experiment is decreased, the distance between adjacent maxima will (1) increase, (2) decrease, (3) remain the same. Explain. (b) If the separation between the two slits is \(0.20 \mathrm{~mm}\) and the adjacent maxima of the interference pattern on a screen \(1.5 \mathrm{~m}\) away from the slits are \(0.45 \mathrm{~cm}\) apart, what is the wavelength and color of the light? (c) If the wavelength is \(550 \mathrm{nm}\), what is the distance between adjacent maxima?

Step-by-Step Solution

Answer

(a) Decrease. (b) 600 nm, orange. (c) 0.0733 cm.

Step 1: Understanding the Double-Slit Formula

In a double-slit experiment, the distance between adjacent maxima (bright fringes) on the screen is given by the formula: \( x = \frac{\lambda D}{d} \), where \( x \) is the fringe spacing, \( \lambda \) is the wavelength of the light, \( D \) is the distance from the slits to the screen, and \( d \) is the separation between the slits.

Step 2: Effect of Decreasing Wavelength

(a) According to the formula \( x = \frac{\lambda D}{d} \), decreasing the wavelength \( \lambda \) results in a decrease in \( x \), the distance between adjacent maxima. Therefore, the distance between adjacent maxima will decrease.

Step 3: Rearrange the Formula to Solve for Wavelength

(b) Using the given values for slit separation \( d = 0.20 \text{ mm} = 0.0002 \text{ m} \), screen distance \( D = 1.5 \text{ m} \), and fringe spacing \( x = 0.45 \text{ cm} = 0.0045 \text{ m} \), rearrange the formula for \( \lambda \):\[ \lambda = \frac{xd}{D} \]

Step 4: Calculate the Wavelength

Substitute the known values into the rearranged formula: \( \lambda = \frac{0.0045 \times 0.0002}{1.5} \), resulting in \( \lambda = 6.0 \times 10^{-7} \text{ m} = 600 \text{ nm} \). The wavelength corresponds to orange light.

Step 5: Calculate New Fringe Spacing for Given Wavelength

(c) If \( \lambda = 550 \text{ nm} = 5.5 \times 10^{-7} \text{ m} \), using the same \( d \) and \( D \), compute \( x \) as follows: \[ x = \frac{(5.5 \times 10^{-7}) \times 0.0002}{1.5} \], which gives \( x = 0.0000733 \text{ m} = 0.0733 \text{ mm} \) or \( 0.0733 \text{ cm} \).

Key Concepts

Wavelength

Wavelength is a key factor in understanding the behavior of waves, particularly in the context of a double-slit experiment. It refers to the distance between consecutive points of equal phase, such as adjacent crests, in a wave. In the double-slit experiment, the wavelength (\(\lambda\)) of light plays a crucial role in determining the position and spacing of the interference pattern seen on a screen.
In this experiment, if the wavelength is decreased, the spacing between adjacent bright fringes (maxima) will decrease. This is because wavelength is directly proportional to fringe spacing (\(x\)) in the formula:
\[ x = \frac{\lambda D}{d} \]
Here:

• \(\lambda\) is the wavelength of the light.
• \(D\) is the distance from the slits to the screen.
• \(d\) is the separation between the slits.

Decreasing the wavelength therefore makes the interference pattern more closely packed on the projection screen.

Interference Pattern

The interference pattern in a double-slit experiment is a visual representation of wave interference. When light waves pass through two slits, they overlap and intersect on the other side, creating a pattern of light and dark areas due to constructive and destructive interference.
Constructive interference occurs when the crests and troughs of the light waves line up perfectly, amplifying the light's intensity and forming bright fringes (maxima). Destructive interference happens when crests meet troughs, canceling each other out and resulting in darkness (minima).
The regular, periodic pattern of light and dark bands is what we refer to as the interference pattern. It can be used to measure the wavelength of light, as any change in the wavelength will affect the spacing and position of these fringes on the screen. The predictability of this pattern demonstrates the wave nature of light.

Fringe Spacing

Fringe spacing in the context of a double-slit experiment refers to the distance between consecutive bright fringes on the screen. This spacing is crucial because it allows us to link the physical setup of the experiment with observable phenomena.
Using the formula, the fringe spacing \(x\) can be determined by:
\[ x = \frac{\lambda D}{d} \]
From this formula, we can see:

• Fringe spacing \(x\) increases with a larger wavelength \(\lambda\).
• Fringe spacing \(x\) increases with a larger distance to the screen \(D\).
• Fringe spacing \(x\) decreases with a greater separation between the slits \(d\).

In experimental setups, measuring fringe spacing allows scientists to backtrack and determine the wavelength of the incoming light based on the setup parameters. This demonstrates the interconnectedness of wave phenomena.

Light Color

The color of light is directly linked to its wavelength. In visible light, different colors appear due to varying wavelengths.

• Violet light has the shortest wavelength, around 400 nm.
• Red light has the longest wavelength, around 700 nm.
• Other colors fall in between, like blue, green, yellow, and orange.

In the step-by-step solution provided, the calculated wavelength was 600 nm, which corresponds to orange light. By determining the wavelength, we can infer the color of the light used in the experiment.
Therefore, understanding how wavelength impacts both the appearance and properties of light offers a hands-on application of these concepts in optics and beyond. This is essential in fields such as telecommunications, imaging, and even in everyday technology like smartphones and TV displays.