Question

For a double-slit experiment, two 1.50 -mm-wide slits are separated by a distance of \(1.00 \mathrm{~mm} .\) The slits are illuminated by laser light with wavelength \(633 \mathrm{nm}\). If a screen is placed \(5.00 \mathrm{~m}\) away from the slits, determine the separation of the bright fringes on the screen.

Short answer

Answer: To find the fringe separation, first, we find the angle between adjacent bright fringes, θ, using the given values: λ = 633 x 10^(-9) m and d = 1.00 x 10^(-3) m. Then, apply the small-angle approximation formula y = L × sin(θ), where L = 5.00 m is the distance between the screen and the slits. By calculating, we will find the distance between the adjacent bright fringes on the screen.

Step-by-step solution

Calculate the angle between the bright fringes

To find the angle between the bright fringes, we can use the double-slit interference formula: \[ n\lambda = d\sin\theta \] where, \(n\) is the order of fringe, \(\lambda\) is the wavelength, \(d\) is the slit separation, and \(\theta\) is the angle between the bright fringes and the central maximum. In our case, adjacent bright fringes would have \(n = 1\) and \(n = 2\). Therefore, we can find the angle between the first two bright fringes. Take the difference, when \(n = 2\) and \(n = 1\), \[ (2\lambda - \lambda) = d\sin\theta \] Simplify the equation: \[ \lambda = d\sin\theta \] Now, we can plug in the values: \(\lambda = 633 \times 10^{-9} \mathrm{m}\) (wavelength), \(d = 1.00 \times 10^{-3}\mathrm{m}\) (slit separation).

Solve for the angle

We can rearrange the equation to solve for the angle \(\theta\): \[ \theta = \mathrm{sin}^{-1} \left(\frac{\lambda}{d}\right) \] Input the values and calculate the angle: \[ \theta = \mathrm{sin}^{-1} \left(\frac{633 \times 10^{-9}\mathrm{m}}{1.00 \times 10^{-3} \mathrm{m}}\right) \]

Calculate the fringe separation

Now that we have the angle, we can calculate the fringe separation on the screen using the small-angle approximation: \[y = L\tan\theta\] where, \( y \) is the fringe separation, \( \theta \) is the angle between the bright fringes and the central maximum, and \( L \) is the distance between the slits and the screen. In our case, the distance is \( L = 5.00 \mathrm{~m} \). With the small-angle approximation, \( \tan \theta \approx \sin \theta\). So, we can rewrite the formula: \[ y = L \sin \theta \] Now, we can plug the values and solve for the fringe separation \(y\): \[ y = 5.00 \mathrm{~m} \times \sin(\theta) \]

Calculate the final result

After finding the angle and putting it into the formula, we can determine the fringe separation (\(y\)). This is the distance between the adjacent bright fringes on the screen located 5.00 meters away from the slits.