{"id":182158,"date":"2025-01-13T13:02:58","date_gmt":"2025-01-13T13:02:58","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=182158"},"modified":"2025-01-13T13:03:00","modified_gmt":"2025-01-13T13:03:00","slug":"you-illuminate-a-slit-with-a-width-of-70-5-um-with-a-light-of-wavelength-727-nm-and-observe-the-resulting-diffraction-pattern-on-a-screen-that-is-situated-2-83-m-from-the-slit","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/13\/you-illuminate-a-slit-with-a-width-of-70-5-um-with-a-light-of-wavelength-727-nm-and-observe-the-resulting-diffraction-pattern-on-a-screen-that-is-situated-2-83-m-from-the-slit\/","title":{"rendered":"You illuminate a slit with a width of 70.5 um with a light of wavelength 727 nm and observe the resulting diffraction pattern on a screen that is situated 2.83 m from the slit"},"content":{"rendered":"\n<p>You illuminate a slit with a width of 70.5 um with a light of wavelength 727 nm and observe the resulting diffraction pattern on a screen that is situated 2.83 m from the slit. What is the width, in centimeters, of the pattern&#8217;s central maximum? width: 8.8 cm Incorrect Light of wavelength 558.0 nm is incident on a narrow slit. The diffraction pattern is viewed on a screen 86.5 cm from the slit. The distance on the screen between the third order minimum and the central maximum is 1.87 cm. What is the width a of the slit in micrometers (um)? a= 5.16 x10-5 um Incorrect<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/01\/image-146.png\" alt=\"\" class=\"wp-image-182159\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To solve these diffraction problems, we need to use the diffraction formula for a single slit, which is given by:<\/p>\n\n\n\n<p>[<br>\\sin(\\theta_m) = \\frac{m \\lambda}{a}<br>]<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( \\theta_m ) is the angle for the m-th order minimum (the first minimum corresponds to ( m = 1 )),<\/li>\n\n\n\n<li>( m ) is the order of the minimum (integer),<\/li>\n\n\n\n<li>( \\lambda ) is the wavelength of the light,<\/li>\n\n\n\n<li>( a ) is the slit width.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Problem 1: Width of the Central Maximum<\/h3>\n\n\n\n<p>In this case, the diffraction pattern&#8217;s central maximum is the region between the first minimum on either side of the central maximum. For a single-slit diffraction pattern, the angle ( \\theta_1 ) for the first minimum (m = 1) is:<\/p>\n\n\n\n<p>[<br>\\sin(\\theta_1) = \\frac{\\lambda}{a}<br>]<\/p>\n\n\n\n<p>To calculate the width of the central maximum, we need to find the distance between the two first-order minima. If ( L ) is the distance from the slit to the screen and ( y_1 ) is the distance from the central maximum to the first minimum on either side, the following relationship holds:<\/p>\n\n\n\n<p>[<br>y_1 = L \\tan(\\theta_1)<br>]<\/p>\n\n\n\n<p>For small angles, ( \\tan(\\theta_1) \\approx \\sin(\\theta_1) ), so:<\/p>\n\n\n\n<p>[<br>y_1 = L \\frac{\\lambda}{a}<br>]<\/p>\n\n\n\n<p>Thus, the total width of the central maximum (from the first minimum on one side to the first minimum on the other side) is:<\/p>\n\n\n\n<p>[<br>\\text{Width} = 2y_1 = 2L \\frac{\\lambda}{a}<br>]<\/p>\n\n\n\n<p>Plugging in the values:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( \\lambda = 727 \\, \\text{nm} = 727 \\times 10^{-9} \\, \\text{m} ),<\/li>\n\n\n\n<li>( a = 70.5 \\, \\mu\\text{m} = 70.5 \\times 10^{-6} \\, \\text{m} ),<\/li>\n\n\n\n<li>( L = 2.83 \\, \\text{m} ),<\/li>\n<\/ul>\n\n\n\n<p>[<br>\\text{Width} = 2 \\times 2.83 \\times \\frac{727 \\times 10^{-9}}{70.5 \\times 10^{-6}} = 0.088 \\, \\text{m} = 8.8 \\, \\text{cm}<br>]<\/p>\n\n\n\n<p>Thus, the width of the central maximum is 8.8 cm, which matches the given answer.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Problem 2: Slit Width<\/h3>\n\n\n\n<p>In this case, we are given the distance between the third-order minimum and the central maximum, which is 1.87 cm. The distance for the third-order minimum, ( y_3 ), is given by:<\/p>\n\n\n\n<p>[<br>y_3 = L \\frac{3 \\lambda}{a}<br>]<\/p>\n\n\n\n<p>From the given values:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( L = 86.5 \\, \\text{cm} = 0.865 \\, \\text{m} ),<\/li>\n\n\n\n<li>( y_3 = 1.87 \\, \\text{cm} = 0.0187 \\, \\text{m} ),<\/li>\n\n\n\n<li>( \\lambda = 558.0 \\, \\text{nm} = 558 \\times 10^{-9} \\, \\text{m} ).<\/li>\n<\/ul>\n\n\n\n<p>Rearranging the formula for ( a ):<\/p>\n\n\n\n<p>[<br>a = \\frac{3 \\lambda L}{y_3}<br>]<\/p>\n\n\n\n<p>Substitute the values:<\/p>\n\n\n\n<p>[<br>a = \\frac{3 \\times 558 \\times 10^{-9} \\times 0.865}{0.0187}<br>]<\/p>\n\n\n\n<p>[<br>a \\approx 5.16 \\times 10^{-5} \\, \\text{m} = 51.6 \\, \\mu\\text{m}<br>]<\/p>\n\n\n\n<p>Thus, the slit width is approximately ( 51.6 \\, \\mu\\text{m} ).<\/p>\n\n\n\n<p>This solution method relies on applying the diffraction equations and knowing how to relate the measured distances to the geometry of the diffraction pattern.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>You illuminate a slit with a width of 70.5 um with a light of wavelength 727 nm and observe the resulting diffraction pattern on a screen that is situated 2.83 m from the slit. What is the width, in centimeters, of the pattern&#8217;s central maximum? width: 8.8 cm Incorrect Light of wavelength 558.0 nm is [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-182158","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182158","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=182158"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182158\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=182158"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=182158"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=182158"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}