Young's Double Slit Experiment

A red laser with a wavelength of $650 \text{ nm}$ ($6.5 \times 10^{-7} \text{ m}$) shines through two slits separated by $0.2 \text{ mm}$ ($2.0 \times 10^{-4} \text{ m}$). The screen is $2.0 \text{ m}$ away. Calculate the distance between the bright fringes.

1. Identify variables: $\lambda = 6.5 \times 10^{-7} \text{ m}$, $L = 2.0 \text{ m}$, $d = 2.0 \times 10^{-4} \text{ m}$. 2. Apply the formula: $\Delta y = \frac{(6.5 \times 10^{-7})(2.0)}{2.0 \times 10^{-4}}$. 3. Calculate: $\Delta y = 0.0065 \text{ m}$ or $6.5 \text{ mm}$.

An interference pattern is formed using green light ($\lambda = 530 \text{ nm}$). If the green light is replaced by violet light ($\lambda = 400 \text{ nm}$) without changing $L$ or $d$, by what factor does the fringe spacing change?

1. Set up a ratio: $\frac{\Delta y_{violet}}{\Delta y_{green}} = \frac{\lambda_{violet}}{\lambda_{green}}$. 2. Substitute values: $\frac{400}{530} \approx 0.75$. 3. Conclusion: The fringes will be $0.75$ times as wide (they will shrink by 25%).

In a double-slit setup, the 3rd order bright fringe ($m=3$) is observed at an angle of $5^{\circ}$. If the slit separation is $0.01 \text{ mm}$, find the wavelength of the light.

1. Use the path difference formula: $d \sin \theta = m\lambda$. 2. Convert $d$ to meters: $0.01 \text{ mm} = 1.0 \times 10^{-5} \text{ m}$. 3. Rearrange for $\lambda$: $\lambda = \frac{d \sin \theta}{m}$. 4. Calculate: $\lambda = \frac{(1.0 \times 10^{-5}) \sin(5^{\circ})}{3} \approx \frac{8.71 \times 10^{-7}}{3} \approx 2.90 \times 10^{-7} \text{ m}$ ($290 \text{ nm}$, which is in the UV spectrum).