{"id":217863,"date":"2025-05-22T15:21:35","date_gmt":"2025-05-22T15:21:35","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=217863"},"modified":"2025-05-22T15:21:37","modified_gmt":"2025-05-22T15:21:37","slug":"consider-the-arrangement-of-point-charges-in-the-figure","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/22\/consider-the-arrangement-of-point-charges-in-the-figure\/","title":{"rendered":"Consider the arrangement of point charges in the figure."},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Consider the arrangement of point charges in the figure.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">a) Determine the direction of the force on q (with q &gt; 0) in the figure, given that qa = qb = +7.50 \u00b5C and qc = qd = -7.50 \u00b5C.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">b) Calculate the magnitude of the force on the charge q in newtons, given that the square is 7 cm on a side and q = 2.75 \u00b5C.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To solve this problem, we consider a square configuration of point charges with a central charge <strong>q<\/strong>. The charges are arranged such that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>qa = qb = +7.50 \u00b5C<\/strong> (positive charges)<\/li>\n\n\n\n<li><strong>qc = qd = -7.50 \u00b5C<\/strong> (negative charges)<\/li>\n\n\n\n<li>Charge <strong>q = +2.75 \u00b5C<\/strong> is located at the <strong>center<\/strong> of the square<\/li>\n\n\n\n<li>The square has sides of <strong>7.00 cm = 0.07 m<\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>a) Direction of the force on q (q &gt; 0)<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Each corner charge exerts an electrostatic force on the central charge <strong>q<\/strong>. Because opposite charges attract and like charges repel, we analyze the net force vectorially:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>qa (+)<\/strong> and <strong>qb (+)<\/strong> (top-left and top-right corners, for example) will <strong>repel<\/strong> the central positive charge <strong>q<\/strong>, pushing it diagonally <strong>away<\/strong> from them.<\/li>\n\n\n\n<li><strong>qc (-)<\/strong> and <strong>qd (-)<\/strong> (bottom-left and bottom-right corners) will <strong>attract<\/strong> the central positive charge <strong>q<\/strong>, pulling it <strong>toward<\/strong> them.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">By symmetry:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The horizontal components of the forces from <strong>qa and qb<\/strong> cancel each other.<\/li>\n\n\n\n<li>The horizontal components of the forces from <strong>qc and qd<\/strong> also cancel.<\/li>\n\n\n\n<li>The <strong>vertical components<\/strong> from <strong>qa and qb<\/strong> both point <strong>downward<\/strong> (repelling q),<\/li>\n\n\n\n<li>The vertical components from <strong>qc and qd<\/strong> also point <strong>downward<\/strong> (attracting q).<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, <strong>all vertical components add<\/strong>, resulting in a <strong>net force directed straight downward<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2705 <strong>Answer for part (a):<\/strong> <strong>The net force on q is directed straight downward.<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>b) Magnitude of the force on q<\/strong><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Each charge at a corner exerts a force on q. Since q is at the center, the distance from each corner to the center is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>r = \\frac{\\text{diagonal}}{2} = \\frac{\\sqrt{2} \\cdot 0.07}{2} \\approx 0.0495 \\text{ m}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using Coulomb\u2019s law for the magnitude of the force between two point charges:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>F = \\frac{k |q_1 q_2|}{r^2}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$k = 8.99 \\times 10^9 \\, \\text{N\u00b7m}^2\/\\text{C}^2$<\/li>\n\n\n\n<li>$q_1 = 7.50 \\times 10^{-6} \\, \\text{C}$<\/li>\n\n\n\n<li>$q_2 = 2.75 \\times 10^{-6} \\, \\text{C}$<\/li>\n\n\n\n<li>$r = 0.0495 \\, \\text{m}$<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>F_{single} = \\frac{(8.99 \\times 10^9)(7.5 \\times 10^{-6})(2.75 \\times 10^{-6})}{(0.0495)^2} \\approx 7.56 \\, \\text{N}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Each force contributes only its vertical component:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>F_{vertical_single} = F_{single} \\cdot \\sin(45^\\circ) = 7.56 \\cdot \\frac{\\sqrt{2}}{2} \\approx 5.35 \\, \\text{N}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There are <strong>four charges<\/strong>, each contributing to a downward component:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>F_{net} = 4 \\cdot 5.35 = \\boxed{21.4 \\, \\text{N downward}}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2705 <strong>Answer for part (b):<\/strong> <strong>The magnitude of the net force on q is approximately 21.4 N, directed downward.<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Summary<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The symmetrical placement of charges leads to cancellation of horizontal components.<\/li>\n\n\n\n<li>Vertical components reinforce due to all pushing\/pulling in the same direction.<\/li>\n\n\n\n<li><\/li>\n\n\n\n<li><\/li>\n\n\n\n<li><\/li>\n\n\n\n<li><\/li>\n\n\n\n<li><\/li>\n\n\n\n<li>The total force is directed <strong>downward<\/strong> with a <strong>magnitude of approximately 21.4 N<\/strong>.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/image-90.png\" alt=\"\" class=\"wp-image-217864\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Consider the arrangement of point charges in the figure. a) Determine the direction of the force on q (with q &gt; 0) in the figure, given that qa = qb = +7.50 \u00b5C and qc = qd = -7.50 \u00b5C. b) Calculate the magnitude of the force on the charge q in newtons, given that [&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-217863","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217863","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=217863"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/217863\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=217863"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=217863"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=217863"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}