{"id":38692,"date":"2025-06-26T11:54:53","date_gmt":"2025-06-26T11:54:53","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=38692"},"modified":"2025-06-26T11:54:55","modified_gmt":"2025-06-26T11:54:55","slug":"current-attempt-in-progress-three-particles-are-fixed-on-an-x-axis","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/current-attempt-in-progress-three-particles-are-fixed-on-an-x-axis\/","title":{"rendered":"Current Attempt in Progress Three particles are fixed on an x axis."},"content":{"rendered":"\n<p>Current Attempt in Progress Three particles are fixed on an x axis. Particle 1 of charge<br>is at x = -a and particle 2 of charge<br>is at x = +a. If their net electrostatic force on particle 3 of charge Q is to be zero, what must be the ratio<br>when particle 3 is at (a) x = +0.544a and (b) x = +2.07a? (a) Number Units (b) Number Units<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>We are given three particles along the x-axis:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Particle 1 has charge q1q_1q1\u200b at position x=\u2212ax = -ax=\u2212a<\/li>\n\n\n\n<li>Particle 2 has charge q2q_2q2\u200b at position x=+ax = +ax=+a<\/li>\n\n\n\n<li>Particle 3 has charge QQQ at position x=x3x = x_3x=x3\u200b, where x3=+0.544ax_3 = +0.544ax3\u200b=+0.544a in part (a), and x3=+2.07ax_3 = +2.07ax3\u200b=+2.07a in part (b)<\/li>\n<\/ul>\n\n\n\n<p>We are asked to find the ratio r=q1q2r = \\frac{q_1}{q_2}r=q2\u200bq1\u200b\u200b such that the <strong>net electrostatic force<\/strong> on particle 3 is <strong>zero<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step-by-Step Process<\/h3>\n\n\n\n<p>The electrostatic force on a charge QQQ from another charge qqq is given by Coulomb\u2019s Law:F=k\u2223qQ\u2223r2F = k \\frac{|qQ|}{r^2}F=kr2\u2223qQ\u2223\u200b<\/p>\n\n\n\n<p>Direction matters: attractive if opposite signs, repulsive if same signs.<\/p>\n\n\n\n<p>Let us assume <strong>all charges are positive<\/strong> for simplicity. The direction of each force will then be away from the other charges.<\/p>\n\n\n\n<p>Let the distance from particle 1 to particle 3 be:r1=x3+ar_1 = x_3 + ar1\u200b=x3\u200b+a<\/p>\n\n\n\n<p>Let the distance from particle 2 to particle 3 be:r2=x3\u2212ar_2 = x_3 &#8211; ar2\u200b=x3\u200b\u2212a<\/p>\n\n\n\n<p>Force from particle 1 on particle 3:F1=kq1Q(x3+a)2F_1 = k \\frac{q_1 Q}{(x_3 + a)^2}F1\u200b=k(x3\u200b+a)2q1\u200bQ\u200b<\/p>\n\n\n\n<p>Force from particle 2 on particle 3:F2=kq2Q(x3\u2212a)2F_2 = k \\frac{q_2 Q}{(x_3 &#8211; a)^2}F2\u200b=k(x3\u200b\u2212a)2q2\u200bQ\u200b<\/p>\n\n\n\n<p>To make net force zero:F1=F2\u21d2q1(x3+a)2=q2(x3\u2212a)2\u21d2q1q2=(x3+ax3\u2212a)2F_1 = F_2 \\Rightarrow \\frac{q_1}{(x_3 + a)^2} = \\frac{q_2}{(x_3 &#8211; a)^2} \\Rightarrow \\frac{q_1}{q_2} = \\left( \\frac{x_3 + a}{x_3 &#8211; a} \\right)^2F1\u200b=F2\u200b\u21d2(x3\u200b+a)2q1\u200b\u200b=(x3\u200b\u2212a)2q2\u200b\u200b\u21d2q2\u200bq1\u200b\u200b=(x3\u200b\u2212ax3\u200b+a\u200b)2<\/p>\n\n\n\n<p>Let us now plug in values.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">(a) When x3=0.544ax_3 = 0.544ax3\u200b=0.544a<\/h3>\n\n\n\n<p>q1q2=(0.544a+a0.544a\u2212a)2=(1.544a\u22120.456a)2=(\u22123.386)2=11.47\\frac{q_1}{q_2} = \\left( \\frac{0.544a + a}{0.544a &#8211; a} \\right)^2 = \\left( \\frac{1.544a}{-0.456a} \\right)^2 = \\left( -3.386 \\right)^2 = 11.47q2\u200bq1\u200b\u200b=(0.544a\u2212a0.544a+a\u200b)2=(\u22120.456a1.544a\u200b)2=(\u22123.386)2=11.47<\/p>\n\n\n\n<p><strong>Answer (a):<\/strong>11.47&nbsp;(unitless)\\boxed{11.47 \\ (\\text{unitless})}11.47&nbsp;(unitless)\u200b<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">(b) When x3=2.07ax_3 = 2.07ax3\u200b=2.07a<\/h3>\n\n\n\n<p>q1q2=(2.07a+a2.07a\u2212a)2=(3.07a1.07a)2=(2.869)2=8.24\\frac{q_1}{q_2} = \\left( \\frac{2.07a + a}{2.07a &#8211; a} \\right)^2 = \\left( \\frac{3.07a}{1.07a} \\right)^2 = \\left( 2.869 \\right)^2 = 8.24q2\u200bq1\u200b\u200b=(2.07a\u2212a2.07a+a\u200b)2=(1.07a3.07a\u200b)2=(2.869)2=8.24<\/p>\n\n\n\n<p><strong>Answer (b):<\/strong>8.24&nbsp;(unitless)\\boxed{8.24 \\ (\\text{unitless})}8.24&nbsp;(unitless)\u200b<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Final Notes:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>These ratios show how the charges must be distributed for particle 3 to experience no net force.<\/li>\n\n\n\n<li>The closer particle 3 is to one charge, the smaller that charge must be to balance the force from the farther one.<\/li>\n\n\n\n<li>The ratios are unitless because we\u2019re comparing charge magnitudes.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1055.jpeg\" alt=\"\" class=\"wp-image-38715\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1055.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1055-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1055-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Current Attempt in Progress Three particles are fixed on an x axis. Particle 1 of chargeis at x = -a and particle 2 of chargeis at x = +a. If their net electrostatic force on particle 3 of charge Q is to be zero, what must be the ratiowhen particle 3 is at (a) x [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-38692","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/38692","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=38692"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/38692\/revisions"}],"predecessor-version":[{"id":38721,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/38692\/revisions\/38721"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=38692"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=38692"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=38692"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}