{"id":6427,"date":"2015-08-28T12:48:08","date_gmt":"2015-08-28T16:48:08","guid":{"rendered":"https:\/\/sciencenotes.org\/?p=6427"},"modified":"2021-05-19T14:53:33","modified_gmt":"2021-05-19T18:53:33","slug":"wavelength-and-energy-example-problem","status":"publish","type":"post","link":"https:\/\/sciencenotes.org\/wavelength-and-energy-example-problem\/","title":{"rendered":"Wavelength and Energy Example Problem"},"content":{"rendered":"\n<div class=\"wp-block-image is-style-default\"><figure class=\"alignright size-medium\"><a href=\"https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/Eye.jpg\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/Eye-300x201.jpg\" alt=\"Human Eye\" class=\"wp-image-6426 lazyload\"\/><noscript><img decoding=\"async\" width=\"300\" height=\"201\" src=\"https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/Eye-300x201.jpg\" alt=\"Human Eye\" class=\"wp-image-6426 lazyload\" srcset=\"https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/Eye.jpg 300w, https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/Eye-150x101.jpg 150w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/noscript><\/a><figcaption>You can use this wavelength and energy example to calculate the number of photons needed to &#8220;see&#8221;. Credit: Petr Nov\u00e1k, Wikipedia<\/figcaption><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">This wavelength and energy example problem will show how to find the energy of a photon from its wavelength.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">First, let&#8217;s look at the physics of the problem. The energy of a single photon of light is dependent on its frequency. This relationship is expressed in the equation<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">E = h\u0192<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where<br>E is the energy of the photon<br>h is Planck&#8217;s constant = 6.626 x 10<sup>-34<\/sup> m<sup>2<\/sup>kg\/s<br>\u0192 is the frequency of the photon<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The <a href=\"https:\/\/sciencenotes.org\/wavelength-to-frequency-calculation-and-equation\/\">wavelength of a photon is related to the frequency<\/a> by the equation<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">c =&nbsp;\u0192\u03bb<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where<br>c is the <a href=\"https:\/\/sciencenotes.org\/what-is-the-speed-of-light\/\">speed of light<\/a> = 3.0 x 10<sup>8<\/sup> m\/s<br>\u0192 is the frequency<br>\u03bb is the wavelength<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Solve this for frequency, and you get<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u0192 = c \/&nbsp;\u03bb<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Substitute this equation into the energy equation and get<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">E = hc \/&nbsp;\u03bb<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">With this equation, you can now find the energy of a photon when the wavelength is known. You can also find the wavelength if the energy of the photon is known.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Wavelength and Energy Example Problem<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Question:<\/strong> The human eye&#8217;s optic nerve needs 2 x 10<sup>-17<\/sup> joules of energy to trigger a series of impulses to signal the brain there is something to see. How many photons of 475 nm blue light is needed to trigger this response?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Solution: We are given the amount of energy needed to trigger the optic nerve and the wavelength of light.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">First, let&#8217;s figure out how much energy is in a single photon of the blue light. We are given the wavelength as 475 nm. Before we go any further, let&#8217;s convert this to meters.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1 nm = 10<sup>-9<\/sup> m<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using this relationship, convert 475 nm to meters<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img decoding=\"async\" class=\"wp-image-86088 lazyload\" style=\"width: 227px;\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/nm2m-new.png\" alt=\"nm to m conversion example\"><noscript><img decoding=\"async\" width=\"227\" height=\"50\" class=\"wp-image-86088 lazyload\" style=\"width: 227px;\" src=\"https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/nm2m-new.png\" alt=\"nm to m conversion example\" srcset=\"https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/nm2m-new.png 454w, https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/nm2m-new-300x66.png 300w, https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/nm2m-new-150x33.png 150w\" sizes=\"(max-width: 227px) 100vw, 227px\" \/><\/noscript><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">x m = 4.75 x 10<sup>-7<\/sup> m<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now we can use the energy formula from above<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">E = hc \/&nbsp;\u03bb<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Plug in the variables<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">E = (6.626 x 10<sup>-34<\/sup> m<sup>2<\/sup>kg\/s)(3 x 10<sup>8<\/sup> m\/s) \/&nbsp;4.75 x 10<sup>-7<\/sup> m<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Solve for E<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">E = 4.18 x 10<sup>-19<\/sup> J<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is the energy of a single photon of 475 nm blue light. We need&nbsp;2 x 10<sup>-17<\/sup> J of energy to begin the process.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">x photons =&nbsp;2 x 10<sup>-17<\/sup> J<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1 photon =&nbsp;4.18 x 10<sup>-19<\/sup> J<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Divide one equation into the other to get<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img decoding=\"async\" class=\"wp-image-86089 lazyload\" style=\"width: 227px;\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/photonratio-new.png\" alt=\"\"><noscript><img decoding=\"async\" width=\"227\" height=\"48\" class=\"wp-image-86089 lazyload\" style=\"width: 227px;\" src=\"https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/photonratio-new.png\" alt=\"\" srcset=\"https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/photonratio-new.png 474w, https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/photonratio-new-300x63.png 300w, https:\/\/sciencenotes.org\/wp-content\/uploads\/2015\/08\/photonratio-new-150x32.png 150w\" sizes=\"(max-width: 227px) 100vw, 227px\" \/><\/noscript><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Solve for x<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">x = 47.8 photons<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since you can&#8217;t have partial photons, we need to round this answer up to the nearest whole photon count. 47 photons isn&#8217;t enough, so one more is needed to get over the threshold energy.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">x = 48 photons<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answer:<\/strong> It takes 48 photons of 475 nm blue light to trigger the optic nerve.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This wavelength and energy example problem will show how to find the energy of a photon from its wavelength.<\/p>\n","protected":false},"author":1406,"featured_media":6426,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[621],"tags":[872,879,1665,322,1668,1667,1664,1666,748,1663],"class_list":["post-6427","post","type-post","status-publish","format-standard","has-post-thumbnail","category-physics-example-problems","tag-conversion","tag-em","tag-electromagnetic-waves","tag-energy-2","tag-meters","tag-nanometers","tag-optics","tag-photon","tag-quantum-mechanics","tag-wavelength","czr-hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.7 - 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