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A diffraction grating has \(4.200 \times 10^{3}\) rulings per centimeter. The screen is \(2.000 \mathrm{~m}\) from the grating. In parts (a) through (e), round each result to four digits, using the rounded values for subsequent calculations. (a) Compute the value of \(d\), the distance between adjacent rulings. Express the answer in meters. (b) Calculate the angle of the second-order maximum made by the \(589.0-\mathrm{nm}\) wavelength of sodium. (c) Find the position of the secondary maximum on the screen. (d) Repeat parts (b) and (c) for the second-order maximum of the \(589.6-\mathrm{nm}\) wavelength of sodium, finding the angle and position on the screen. (e) What is the distance between the two secondary maxima on the screen? (f) Now find the distance between the two secondary maxima without rounding, carrying all digits in calculator memory. How many significant digits are in agreement? What can you conclude about rounding and the use of intermediate answers in this case?

Step by step solution

01

Find distance between rulings

To find the distance between adjacent rulings, use the formula \(d = 1/N\), where N is the number of rulings per meter. Since we have a number of rulings per centimeter, first convert it to per meter by multiplying it by \(100\). So, N = \(4.200 * 10^{3} rulings/cm * 100 cm/m = 4.200 * 10^{5}\) rulings/m. Now, calculate \(d = 1/N = 1/(4.200 * 10^{5}) m = 2.381 * 10^{-6} m\).

02

Calculate the angle of second-order maximum

To find this angle, use the formula of diffraction grating: \(d * sin(\Theta) = m * λ\), where m is the order of maximum, λ is the wavelength, and \(\Theta\) is the angle we are going to find. Plugging in the values we get: \(2.381 * 10^{-6}\ m * sin(\Theta) = 2 * 589.0 * 10^{-9} m\). Solving for sin(\(\Theta\)) we get sin(\(\Theta\) = 0.4945, thus \(\Theta = arcsin(0.4945) = 30.01°\).

03

Compute the position of secondary maximum on the screen

The position of the secondary maximum on the screen is computed using \(L * tan(\Theta)\) where \(L = 2.000\) m and \(\Theta = 30.01°\). From this we get \(L * tan(\Theta) = 1.155\) m.

04

Repeat for different wavelength

We find the angle and position on the the screen for the second-order maximum of the \(589.6-nm\) wavelength by repeating steps 2 and 3 but replacing the wavelength in the calculations with \(589.6-nm\).

05

Compute the distance between secondary maxima

The distance between the maxima on the screen resulting from the two slightly different wavelengths of sodium can be found by subtracting the results attained.

06

Repeat without rounding

By retaining all the decimal places in the calculator, perform the above calculations again to determine the distance between the maxima. The results obtained can be compared with those of Step 5 to understand the effects of rounding in calculations.

07

Analyze the effect of rounding

Finally, compare the number of significant digits in agreement in both cases to conclude the effect of rounding and using intermediate answers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wavelength

Wavelength is a fundamental concept in the study of wave phenomena, characterizing the distance between consecutive peaks (or troughs) of a wave. In the context of a diffraction grating, wavelengths such as 589.0 nm and 589.6 nm are crucial as they determine how light bends when it interacts with a grating.

A shorter wavelength means the lightwave will undergo a smaller angle of diffraction, while a longer wavelength results in a greater diffraction angle. Accurate measurement and usage of wavelength values are essential to predicting the patterns seen in diffraction experiments accurately.

Diffraction Angle

The diffraction angle (\(\Theta\)) is the angle at which waves emerge from a diffraction grating relative to the original direction of the incoming wave. It is given by the grating equation:\[d \cdot \sin(\Theta) = m \cdot \lambda\]where \(d\) is the spacing between rulings on the grating, \(m\) is the order of the diffraction (in this exercise, second-order which means \(m = 2\)), and \(\lambda\) is the wavelength of the light.

• For example, with a wavelength \(\lambda = 589.0 \text{ nm}\), the diffraction angle can be calculated to be approximately \(30.01°\).
• The angle increases as the wavelength increases, or as the order of maximum \(m\) increases.

Therefore, understanding the diffraction angle is crucial for determining the precise path that light will follow when it passes through a grating.

Secondary Maximum

A secondary maximum is the intensity peak observed on a screen when light waves constructively interfere after passing through a diffraction grating at specific angles. The location of these maxima can be determined using the trigonometric relation:\[\text{Position} = L \cdot \tan(\Theta)\]where \(L\) is the distance from the grating to the screen and \(\Theta\) is the diffraction angle.

In this exercise:

• With \(L = 2.000 \text{ m}\) and \(\Theta = 30.01°\), the secondary maximum's position is approximately \(1.155 \text{ m}\) from the central maximum.
• Calculating for different wavelengths yields different positions, highlighting how even small wavelength changes can lead to noticeable shifts in interference patterns.

These shifts are instrumental in applications like spectroscopy, where various wavelengths must be accurately resolved.

Significant Digits

Significant digits play a crucial role in scientific calculations, especially when multiple sequential calculations are involved, as seen in this exercise. They determine the precision of the results and help prevent cumulative errors from impacting the final outcomes.

Initially, rounding values to four significant digits allows for consistent precision across all steps. However:

• In Step 6 of the provided exercise, calculations were performed without rounding to assess how rounding affects results.
  The agreement between outcomes from rounded and unrounded calculations sheds light on how much precision might be lost.
• Step 7 discusses comparing significant digits in both methods to illustrate rounding's impact and guide on when retaining more precise figures might be crucial.
  This importance magnifies in experiments where precision directly influences conclusions.

Overall, understanding significant digits helps maintain consistent accuracy and reliability in scientific measurements and calculations.