You set your slit spacing at 1.13 mm and place your screen 8.75 m from the slits. Then, you illuminate the slits and find that the tenth bright fringe is 4.95 cm away from the central bright fringe...

Question

You set your slit spacing at 1.13 mm and place your screen 8.75 m from the slits. Then, you illuminate the slits and find that the tenth bright fringe is 4.95 cm away from the central bright fringe (counted as 0). What is the wavelength of the light in nanometers?

Answer 1

To find the wavelength of the light, we can use the formula for the angular positions of fringes in a double-slit experiment:

\[θ = \frac{λ}{d}\]

Where:

- \(θ\) is the angle between the central maximum and the fringe.

- \(λ\) is the wavelength of the light (in meters).

- \(d\) is the distance between the slits (in meters).

Given the slit spacing \(d = 1.13 \, \text{mm}\), we need to convert it to meters by dividing by 1000:

\[d = 1.13 \, \text{mm} = 0.00113 \, \text{m}\]

Now, we can find the angular position (\(θ_{10}\)) for the tenth bright fringe:

\[θ_{10} = \frac{4.95 \, \text{cm}}{8.75 \, \text{m}}\]

First, convert the centimeters to meters:

\[θ_{10} = \frac{0.0495 \, \text{m}}{8.75 \, \text{m}}\]

Now, calculate \(θ_{10}\):

\[θ_{10} ≈ 0.0056571 \, \text{radians}\]

Now, use the formula to find the wavelength (\(λ\)):

\[λ = θ_{10} \cdot d\]

Substitute the values:

\[λ = 0.0056571 \, \text{radians} \cdot 0.00113 \, \text{m}\]

Now, calculate \(λ\) in meters:

\[λ ≈ 0.000006408 \, \text{m}\]

Finally, convert meters to nanometers (1 meter = 1,000,000,000 nanometers):

\[λ ≈ 6.408 \, \text{nm}\]

So, the wavelength of the light is approximately \(6.408 \, \text{nanometers}\).