A solenoid of length 0.700 m having a circular cross-section of radius 5.00 cm stores 6.00 \u00b5J of energy when a 0.400-A current runs through it. What is the winding density of the solenoid? (\u03bc 0 = 4\u03c0 \u00d7 10-7 T \u00b7 m\/A)<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n To find the winding density (number of turns per unit length) of the solenoid, we can use the energy stored in the solenoid and the formula for the magnetic energy stored in an inductor.<\/p>\n\n\n\n The energy ( U ) stored in an inductor is given by:<\/p>\n\n\n\n [ where:<\/p>\n\n\n\n Rearranging the equation to solve for inductance ( L ):<\/p>\n\n\n\n [ Substituting in the values:<\/p>\n\n\n\n [ Next, the inductance ( L ) of a solenoid is also given by the formula:<\/p>\n\n\n\n [ where:<\/p>\n\n\n\n The cross-sectional area ( A ) of the solenoid can be calculated using the radius ( r ):<\/p>\n\n\n\n [ Substituting the values into the inductance formula:<\/p>\n\n\n\n [ We can solve for ( n ):<\/p>\n\n\n\n [ Substituting the known values:<\/p>\n\n\n\n [ Calculating ( \\mu_0 A l ):<\/p>\n\n\n\n [ Finally, substituting into the equation for ( n ):<\/p>\n\n\n\n [ Thus, the winding density of the solenoid is approximately 40,646 turns\/m<\/strong>.<\/p>\n\n\n\n The winding density, or turns density, represents how many coils of wire are wrapped around the solenoid per unit length. The energy stored in the magnetic field of the solenoid depends on both the current flowing through the coils and the arrangement of those coils (winding density). Using the principles of electromagnetism, particularly the relationships between energy, inductance, and magnetic field, we can effectively determine how tightly the solenoid is wound based on its dimensions and the current it carries.<\/p>\n","protected":false},"excerpt":{"rendered":" A solenoid of length 0.700 m having a circular cross-section of radius 5.00 cm stores 6.00 \u00b5J of energy when a 0.400-A current runs through it. What is the winding density of the solenoid? (\u03bc 0 = 4\u03c0 \u00d7 10-7 T \u00b7 m\/A) The Correct Answer and Explanation is : To find the winding density […]<\/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-153960","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/153960","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=153960"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/153960\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=153960"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=153960"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=153960"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
U = \\frac{1}{2} L I^2
]<\/p>\n\n\n\n\n
L = \\frac{2U}{I^2}
]<\/p>\n\n\n\n
L = \\frac{2 \\times 6.00 \\times 10^{-6} \\text{ J}}{(0.400 \\text{ A})^2} = \\frac{12.00 \\times 10^{-6}}{0.160} = 7.50 \\times 10^{-5} \\text{ H}
]<\/p>\n\n\n\n
L = \\mu_0 n A l
]<\/p>\n\n\n\n\n
A = \\pi r^2 = \\pi (0.05)^2 = \\pi (0.0025) = 0.007854 \\text{ m}^2
]<\/p>\n\n\n\n
L = \\mu_0 n A l
]<\/p>\n\n\n\n
n = \\frac{L}{\\mu_0 A l}
]<\/p>\n\n\n\n
n = \\frac{7.50 \\times 10^{-5} \\text{ H}}{(4\\pi \\times 10^{-7} \\text{ T\u00b7m\/A}) (0.007854 \\text{ m}^2) (0.700 \\text{ m})}
]<\/p>\n\n\n\n
\\mu_0 A l = (4\\pi \\times 10^{-7}) (0.007854) (0.700) = 1.847 \\times 10^{-9}
]<\/p>\n\n\n\n
n = \\frac{7.50 \\times 10^{-5}}{1.847 \\times 10^{-9}} \\approx 40645.6 \\text{ turns\/m}
]<\/p>\n\n\n\nExplanation<\/h3>\n\n\n\n