{"id":2395,"date":"2018-05-04T17:50:59","date_gmt":"2018-05-04T09:50:59","guid":{"rendered":"https:\/\/gurumuda.net\/physics\/?p=2395"},"modified":"2023-08-06T15:25:37","modified_gmt":"2023-08-06T15:25:37","slug":"diffraction-grating-problems-and-solutions","status":"publish","type":"post","link":"https:\/\/gurumuda.net\/physics\/diffraction-grating-problems-and-solutions.htm","title":{"rendered":"Diffraction grating \u2013 problems and solutions","gt_translate_keys":[{"key":"rendered","format":"text"}]},"content":{"rendered":"

Diffraction grating \u2013 problems and solutions<\/span><\/p>\n

1. A grating<\/a> containing 4000 slits per centimeter is illuminated with a monochromatic <\/span>light and produces the second-order bright line at a 30\u00b0 angle. What is the wavelength<\/a> of the light used? (1 \u00c5 = 10-10<\/sup> m)<\/span><\/p>\n

Known :<\/u><\/span><\/p>\n

The distance between slits (d) = 1 \/ (4000 slits \/ cm) = 0.00025 cm = 2.5 x 10-4<\/sup> cm = 2.5 x 10-6<\/sup> meters<\/span><\/p>\n

Order (n) = 2<\/span><\/p>\n

Sin 30o<\/sup> = 0.5<\/span><\/p>\n

1 \u00c5 = 10-10<\/sup> m<\/span><\/p>\n

Wanted:<\/u> Wavelength of the light (\u03bb)<\/span><\/p>\n

Solution :<\/u><\/span><\/p>\n

\"Diffraction<\/span><\/p>\n

2. A monochromatic <\/span>light with a wavelength of 2.5×10-7 <\/sup>m strikes a grating containing 10,000 slits\/cm. Determine the angular positions of the second-order bright line.<\/span><\/p>\n

Known :<\/u><\/span><\/p>\n

The distance between slits (d) = 1 \/ (10,000 slits \/ cm) = 0.0001 cm = 1 x 10-4<\/sup> cm = 1 x 10-6<\/sup> m<\/span><\/p>\n

Order (n) = 2<\/span><\/p>\n

Wavelength (\u03bb) = 2.5 x 10-7 <\/sup>m<\/span><\/p>\n

Wanted :<\/u> Angle (\u03b8)<\/span><\/p>\n

Solution :<\/u><\/span><\/p>\n

\"Diffraction<\/span><\/p>\n

3. A monochromatic <\/span>light with wavelength of 5.10-7<\/sup> m strikes a grating. Distance between slit and the viewing screen<\/span> is 2 m, distance between the <\/span>third<\/span>-order fringe and the central fringe<\/span> is 150 cm. Determine distance between slits.<\/span><\/p>\n

Known :<\/u><\/span><\/p>\n

Wavelength (\u03bb) = 5 x 10-7<\/sup> m<\/span><\/p>\n

l = 2 m<\/span><\/p>\n

n = 3<\/span><\/p>\n

y = 150 cm = 1.5 m<\/span><\/p>\n

Wanted :<\/u> distance between slits<\/span><\/p>\n

Solution :<\/u><\/span><\/p>\n

sin \u03b8 \u2248 tan \u03b8 = y \/ l = 1.5 \/ 2 = 0.75 = 75 x 10-2<\/sup><\/span><\/p>\n

Distance between slits :<\/span><\/p>\n

d sin \u03b8 = n \u03bb<\/span><\/p>\n

d (75.10-2<\/sup>) = (3)(5.10-7<\/sup>)<\/span><\/p>\n

d (75.10-2<\/sup>) = 15.10-7<\/sup><\/span><\/p>\n

d = (15.10-7<\/sup>) \/ (75.10-2<\/sup>) <\/span><\/p>\n

d = (15\/75).10-5<\/sup><\/span><\/p>\n

d = (1\/5)(10-5<\/sup>) m<\/span><\/p>\n

d = (0.2)(10-5<\/sup>) m<\/span><\/p>\n

d = 2.10-6 <\/sup>m<\/span><\/p>\n

4. A monochromatic <\/span>light with wavelength of 500 nm (1 nano = 10-9<\/sup>) strikes a grating and produces the fourth-order bright line at an 30\u00b0 angle. Determine the number of slits per centimeter.<\/span><\/p>\n

Known :<\/u><\/span><\/p>\n

Wavelength (\u03bb) = 500.10-9<\/sup> m = 5.10-7<\/sup> m<\/span><\/p>\n

\u03b8 = 30o<\/sup><\/span><\/p>\n

n = 4<\/span><\/p>\n

Wanted :<\/u> number of slits per centimeter<\/span><\/p>\n

Solution :<\/u><\/span><\/p>\n

Distance between slits :<\/span><\/p>\n

d sin \u03b8 = n \u03bb<\/span><\/p>\n

d (sin 30o<\/sup>) = (4)(5.10-7<\/sup>)<\/span><\/p>\n

d (0.5) = 20.10-7<\/sup><\/span><\/p>\n

d = (20.10-7<\/sup>) \/ 0.5 <\/span><\/p>\n

d = 40.10-7<\/sup><\/span><\/p>\n

d = 4.10-6 <\/sup>m<\/span><\/p>\n

Number of slits per centimeter :<\/span><\/p>\n

x = 1 \/ 4.10-6 <\/sup>m<\/span><\/p>\n

x = 0.25.106<\/sup> \/ m<\/span><\/p>\n

x = 0.25.106<\/sup> \/ 102<\/sup> cm<\/span><\/p>\n

x = 0.25.104<\/sup> \/ cm<\/span><\/p>\n

x = 25.102<\/sup> \/ cm<\/span><\/p>\n

x = 2500 \/ cm<\/span><\/p>\n

5. A monochromatic <\/span>light with wavelength of 500 nm (1 nm = 10-9<\/sup> m) strikes a grating and produces the second-order bright line at an 30\u00b0 angle. Determine the number of slits per centimeter.<\/span><\/p>\n

Known :<\/u><\/span><\/p>\n

Wavelength (\u03bb) = 500.10-9<\/sup> m = 5.10-7<\/sup> m<\/span><\/p>\n

\u03b8 = 30o<\/sup><\/span><\/p>\n

n = 2<\/span><\/p>\n

Wanted :<\/u> number of slits per centimeter<\/span><\/p>\n

Solution :<\/u><\/span><\/p>\n

Distance between slits :<\/span><\/p>\n

d sin \u03b8 = n \u03bb<\/span><\/p>\n

d (sin 30o<\/sup>) = (2)(5.10-7<\/sup>)<\/span><\/p>\n

d (0.5) = 10.10-7<\/sup><\/span><\/p>\n

d = (10.10-7<\/sup>) \/ 0.5 <\/span><\/p>\n

d = 20.10-7<\/sup><\/span><\/p>\n

d = 2.10-6 <\/sup>m<\/span><\/p>\n

Number of slits per centimeter :<\/span><\/p>\n

x = 1 \/ d<\/span><\/p>\n

x = 1 \/ 2.10-6 <\/sup>m<\/span><\/p>\n

x = 0.5.106<\/sup> \/ 1 m<\/span><\/p>\n

x = 0.5.106<\/sup> \/ 102<\/sup> cm<\/span><\/p>\n

x = 0.5.104<\/sup> \/ cm<\/span><\/p>\n

x = 5.103<\/sup> \/ cm<\/span><\/p>\n

x = 5000 \/ cm<\/span><\/p>\n

\n
\n
\n
\n
\n
\n
\n
\n
\n
\n

20 conceptual questions and answers about diffraction grating:<\/span><\/p>\n

1. Question:<\/strong> What is a diffraction grating?<\/span><\/p>\n

Answer:<\/strong> A diffraction grating is an optical component with a periodic structure, which splits and diffracts light into several beams traveling in different directions.<\/span><\/p>\n

2. Question:<\/strong> How does diffraction grating differ from a double slit?<\/span><\/p>\n

Answer:<\/strong> While both cause diffraction, a grating has many more slits (or lines) per unit length, producing more diffracted beams and clearer spectral separation.<\/span><\/p>\n

3. Question:<\/strong> Why are the spacings between grating lines crucial?<\/span><\/p>\n

Answer:<\/strong> The spacings determine the angles at which different wavelengths of light are diffracted, thereby affecting the resolution and clarity of the spectrum produced.<\/span><\/p>\n

4. Question:<\/strong> How is the diffraction angle related to the wavelength of light?<\/span><\/p>\n

Answer:<\/strong> The relationship is given by the grating equation: \ufffd\ufffd=\ufffdsin\u2061\ufffd<\/span>m\u03bb<\/span>=<\/span><\/span>d<\/span>sin<\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>, where \ufffd<\/span>m<\/span><\/span><\/span><\/span><\/span> is the order of diffraction, \ufffd<\/span>\u03bb<\/span><\/span><\/span><\/span><\/span> is the wavelength, \ufffd<\/span>d<\/span><\/span><\/span><\/span><\/span> is the grating spacing, and \ufffd<\/span>\u03b8<\/span><\/span><\/span><\/span><\/span> is the diffraction angle.<\/span><\/p>\n

6. Question:<\/strong> Why do different colors\/wavelengths diffract at different angles?<\/span><\/p>\n

Answer:<\/strong> Because the diffraction angle (\ufffd<\/span>\u03b8<\/span><\/span><\/span><\/span><\/span>) is directly proportional to the wavelength (\ufffd<\/span>\u03bb<\/span><\/span><\/span><\/span><\/span>), longer wavelengths (like red) diffract more than shorter ones (like blue).<\/span><\/p>\n

7. Question:<\/strong> How does increasing the number of slits (or lines) in a diffraction grating affect the pattern?<\/span><\/p>\n

Answer:<\/strong> Increasing the number of slits sharpens the diffraction maxima, enhancing the resolution of the spectrum.<\/span><\/p>\n

8. Question:<\/strong> What is meant by the resolving power of a grating?<\/span><\/p>\n

Answer:<\/strong> Resolving power is the grating’s ability to distinguish between two closely spaced wavelengths. It increases with more lines and higher diffraction orders.<\/span><\/p>\n

9. Question:<\/strong> Can diffraction gratings be used with other types of waves besides light?<\/span><\/p>\n

Answer:<\/strong> Yes, they can be used with any wave phenomenon, including sound waves and X-rays.<\/span><\/p>\n

10. Question:<\/strong> How is a transmission grating different from a reflection grating?<\/span><\/p>\n

Answer:<\/strong> In a transmission grating, light passes through the grating material, whereas in a reflection grating, light is reflected off the grating surface.<\/span><\/p>\n

11. Question:<\/strong> What are blazed gratings?<\/span><\/p>\n

Answer:<\/strong> Blazed gratings are reflection gratings where the grooves are angled (or blazed) to direct most of the energy into a specific diffraction order, optimizing efficiency.<\/span><\/p>\n

12. Question:<\/strong> Why is a grating’s groove density (lines per mm) significant?<\/span><\/p>\n

Answer:<\/strong> Groove density determines the grating’s dispersion and resolution. Higher groove densities generally lead to higher resolutions.<\/span><\/p>\n

13. Question:<\/strong> How does a grating produce a spectrum?<\/span><\/p>\n

Answer:<\/strong> When white light interacts with a grating, each wavelength diffracts at a unique angle, spreading them out spatially to produce a spectrum.<\/span><\/p>\n

14. Question:<\/strong> Why might higher orders of diffraction overlap with one another?<\/span><\/p>\n

Answer:<\/strong> For different wavelengths, higher orders can diffract at angles that match lower orders of other wavelengths, causing overlapping spectra.<\/span><\/p>\n

15. Question:<\/strong> Can a grating be used to study non-visible wavelengths?<\/span><\/p>\n

Answer:<\/strong> Yes, gratings can be designed for different parts of the electromagnetic spectrum, including ultraviolet and infrared.<\/span><\/p>\n

16. Question:<\/strong> What role does grating material play in diffraction patterns?<\/span><\/p>\n

Answer:<\/strong> Material properties can affect the efficiency of different diffraction orders and determine the wavelength range for which the grating is optimized.<\/span><\/p>\n

17. Question:<\/strong> How is a holographic diffraction grating produced?<\/span><\/p>\n

Answer:<\/strong> Holographic gratings are created using interference patterns from laser light, rather than mechanically ruling grooves, resulting in less stray light and higher purity spectra.<\/span><\/p>\n

18. Question:<\/strong> Why is the zero-order maximum usually much brighter?<\/span><\/p>\n

Answer:<\/strong> The zero-order maximum corresponds to undiffracted light. Since it doesn’t undergo as much spreading as diffracted orders, its intensity is generally higher.<\/span><\/p>\n

19. Question:<\/strong> Can gratings work with polarized light?<\/span><\/p>\n

Answer:<\/strong> Yes, the diffraction efficiency and pattern can vary based on the polarization state of the incident light.<\/span><\/p>\n

20. Question:<\/strong> Why do gratings have advantages over prisms for spectroscopy?<\/span><\/p>\n

Answer:<\/strong> Gratings can offer better resolution, more linear dispersion of wavelengths, and are often more efficient at spreading out light across the spectrum compared to prisms.<\/span><\/p>\n

Diffraction gratings are essential in optical research and applications, especially in spectroscopy, allowing for detailed study of light’s spectral components.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"excerpt":{"rendered":"

Diffraction grating \u2013 problems and solutions 1. A grating containing 4000 slits per centimeter is illuminated with a monochromatic light and produces the second-order bright line at a 30\u00b0 angle. What is the wavelength of the light used? (1 \u00c5 = 10-10 m) Known : The distance between slits (d) = 1 \/ (4000 slits … Read more<\/a><\/p>\n","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_seopress_titles_title":"","_seopress_titles_desc":"","_seopress_robots_index":"","_seopress_robots_follow":"","_seopress_robots_imageindex":"","_seopress_robots_snippet":"","_seopress_robots_primary_cat":"","_seopress_robots_breadcrumbs":"","_seopress_robots_freeze_modified_date":"","_seopress_robots_custom_modified_date":"","_seopress_robots_canonical":"","_seopress_social_fb_title":"","_seopress_social_fb_desc":"","_seopress_social_fb_img":"","_seopress_social_fb_img_attachment_id":0,"_seopress_social_fb_img_width":0,"_seopress_social_fb_img_height":0,"_seopress_social_twitter_title":"","_seopress_social_twitter_desc":"","_seopress_social_twitter_img":"","_seopress_social_twitter_img_attachment_id":0,"_seopress_social_twitter_img_width":0,"_seopress_social_twitter_img_height":0,"_seopress_redirections_value":"","_seopress_redirections_enabled":"","_seopress_redirections_enabled_regex":"","_seopress_redirections_logged_status":"","_seopress_redirections_param":"","_seopress_redirections_type":0,"_seopress_analysis_target_kw":"Diffraction grating \u2013 problems and solutions","_seopress_news_disabled":"","_seopress_video_disabled":"","_seopress_video":[],"_seopress_pro_schemas_manual":[],"_seopress_pro_rich_snippets_disable_all":"","_seopress_pro_rich_snippets_disable":[],"_seopress_pro_schemas":[],"footnotes":""},"categories":[3],"tags":[],"class_list":["post-2395","post","type-post","status-publish","format-standard","hentry","category-solved-problems-in-basic-physics"],"gt_translate_keys":[{"key":"link","format":"url"}],"_links":{"self":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts\/2395","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/comments?post=2395"}],"version-history":[{"count":2,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts\/2395\/revisions"}],"predecessor-version":[{"id":8573,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts\/2395\/revisions\/8573"}],"wp:attachment":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/media?parent=2395"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/categories?post=2395"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/tags?post=2395"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}