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?

Step by step solution

01

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: \(\sin \theta_{min} = \frac{\lambda}{a}\)

02

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: \(\sin \theta_1 = \frac{\lambda}{d}\)

03

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: \(\frac{\sin \theta_{min}}{\sin \theta_1} = \frac{d}{a}\)

04

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 \(d = \frac{\lambda}{\sin \theta_1}\)

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