Question

The irradiance pattern observed in a two-slit interference-diffraction experiment is presented in the figure. The red line represents the actual intensity measured as a function of angle, while the green line represents the envelope of the interference patterns. a) Determine the slit width \(a\) in terms of the wavelength \(\lambda\) of the light used in the experiment. b) Determine the center-to-center slit separation \(d\) in terms of the wavelength \(\lambda\). c) Using the information in the graph, determine the ratio of slit width \(a\) to the center-to-center separation between the slits, \(d\). d) Can you calculate the wavelength of light, actual slit separation, and slit width?

Short answer

Using the given step-by-step solution to create a short answer: To solve a two-slit interference-diffraction exercise, we first determine the slit width 'a' and center-to-center slit separation 'd' in terms of the wavelength 'λ' using formulas related to diffraction and interference patterns. Then, we calculate the ratio of slit width 'a' to the center-to-center separation 'd' using values of the angles from the graph. Finally, with the additional information like the fringe spacing and screen distance, we can calculate the actual values of the wavelength, slit separation, and slit width.

Step-by-step solution

Determine the slit width 'a' in terms of the wavelength 'λ'

To determine the slit width 'a' in terms of the wavelength 'λ,' we need to consider the diffraction pattern. The spacing between the envelopes (minima) of the diffraction pattern is given by the formula: \(\sin \theta = \frac{m \lambda}{a}\) where \(\theta\) is the angle from the central maximum, \(m\) is the order of the pattern, and \(a\) is the slit width. The spacing between adjacent envelopes is given when \(m = 1\). So we have: \(\sin \theta_{min} = \frac{\lambda}{a}\)

Determine the center-to-center slit separation 'd' in terms of the wavelength 'λ'

To determine the center-to-center slit separation 'd' in terms of the wavelength 'λ,' we need to consider the interference pattern. The spacing between the bright fringes of the interference pattern is given by the formula: \(\sin \theta_n = \frac{n \lambda}{d}\) where \(\theta_n\) is the angle from the central maximum, \(n\) is the order of the pattern, and \(d\) is the center-to-center slit separation. We use \(n=1\) to find the spacing between adjacent bright fringes: \(\sin \theta_1 = \frac{\lambda}{d}\)

Determine the ratio of slit width 'a' to the center-to-center separation 'd'

From the previous steps, we have \(\sin \theta_{min} = \frac{\lambda}{a}\) and \(\sin \theta_1 = \frac{\lambda}{d}\). From the graph provided in the problem, we can read the values of \(\theta_{min}\) and \(\theta_1\). Then divide these two equations: \(\frac{\sin \theta_{min}}{\sin \theta_1} = \frac{\frac{\lambda}{a}}{\frac{\lambda}{d}}\) As we can see, the wavelength 'λ' cancels out in this equation, so we're left with: \(\frac{\sin \theta_{min}}{\sin \theta_1} = \frac{d}{a}\)

Calculate the wavelength, actual slit separation, and slit width

To find the actual values of the wavelength, slit separation, and slit width, we need some additional information, usually provided by the experimental setup or measurement. For instance, wavelength can be determined if the fringes spacing in the pattern and the distance of the screen from the slits are known. With actual values of any one of these three parameters, we can use the previously derived equations (Step 1 & 2) to find the remaining parameters: \(a = \frac{\lambda}{\sin \theta_{min}}\) and \(d = \frac{\lambda}{\sin \theta_1}\)

Key concepts

Diffraction Pattern

A diffraction pattern is a crucial phenomenon observed in experiments involving waves, such as light passing through a barrier with slits. In the context of a two-slit interference-diffraction experiment, it refers to the intensity pattern formed by the light waves spreading out after passing through the slits. These patterns are characterized by alternating dark and bright bands that result from the behavior of light waves bending around the edges of the slits.

The light waves interfere with each other, creating a distinct pattern on the screen behind the slits. The key part of understanding diffraction patterns is recognizing how the waves add up to form regions of high and low intensity based on where they meet either in-phase or out-of-phase. Importantly, the spacing between these envelopes or minima in a diffraction pattern can be explained using the equation: \[ \sin \theta = \frac{m \lambda}{a} \] where \( \theta \) is the angle related to the central maximum, \( m \) is the order number, \( \lambda \) is the wavelength of light, and \( a \) is the slit width.

Slit Width

The slit width, denoted by \( a \), plays a significant role in forming diffraction patterns. It refers to the physical width of each slit in the barrier. A smaller slit width causes the diffraction pattern to spread more, with greater spacing between its minima. This is because the diffraction effect is more pronounced when light passes through a narrower gap. On the other hand, a larger slit width results in a more concentrated pattern.

Mathematically, to find the slit width in terms of light wavelength \( \lambda \), we use the equation related to the diffraction minima: \[ \sin \theta_{min} = \frac{\lambda}{a} \] Here, \( \theta_{min} \) is the angle at which the minima occur. This formula helps us determine how wide the slit needs to be for specific light wavelengths or diffraction angles used during an experiment.

Center-to-Center Slit Separation

Center-to-center slit separation is another critical factor impacting interference patterns. It is represented by the symbol \( d \) and signifies the distance between the centers of two consecutive slits. This parameter influences how closely or widely spaced the bright fringes appear in an interference pattern.

The formula that describes the relationship between the slit separation \( d \) and the light wavelength \( \lambda \) is: \[ \sin \theta_n = \frac{n \lambda}{d} \] where \( \theta_n \) is the angle of the nth order maximum, and \( n \) is an integer indicating the order of the fringe. By controlling the value of \( d \), one can manipulate the distances between bright and dark regions seen in interference, thereby affecting the overall appearance of the pattern.

Interference Pattern

Interference patterns result from the superposition of light waves, and they are visible as alternating light and dark bands on a screen. These patterns arise because waves emitted from each slit overlap and interact with each other. Where the waves meet in phase, they form bright bands (constructive interference), and where they meet out of phase, they result in dark bands (destructive interference).

The interference pattern is distinct from the diffraction pattern, though they often occur together, especially in experiments with double slits. The involvement of both creates an overall pattern that can be difficult to distinguish without understanding each component's contribution. The bright fringe locations are dictated by the principle: \[ \sin \theta_n = \frac{n \lambda}{d} \] Recognizing these patterns is essential for interpreting experimental results in optics and understanding the underlying wave nature of light.