{"id":988,"date":"2024-07-23T10:34:00","date_gmt":"2024-07-23T10:34:00","guid":{"rendered":"https:\/\/www.meniit.com\/study-material\/?p=988"},"modified":"2024-10-04T05:51:35","modified_gmt":"2024-10-04T05:51:35","slug":"amperes-circuital-law","status":"publish","type":"post","link":"https:\/\/www.meniit.com\/study-material\/jee\/class-xith\/11th-physics\/amperes-circuital-law","title":{"rendered":"Ampere\u2019s Circuital Law"},"content":{"rendered":"

This law is useful in finding the magnetic field due to currents under certain conditions of symmetry. Consider a closed plane curve enclosing some current-carrying conductors.<\/p>\n

\"current-carrying-conductors\"<\/p>\n

Where I<\/em> = total current (algebraic sum) crossing the area.<\/p>\n

\"sign-of-current\"<\/p>\n

As a simple application of this law, we can derive the magnetic induction due to a long straight wire carrying current I<\/em>.<\/p>\n

Suppose the magnetic induction at point P, distant R from the wire is required.<\/p>\n

Draw the circle through P with centre O and radius R.<\/p>\n

\"sign-of-current1\"<\/p>\n

\n

Example :<\/strong><\/h4>\n

Write equation for Ampere\u2019s circuital law for the Amperian loop shown (traversed in the direction shown by arrow marks put on it).<\/p>\n

<\/p>\n<\/div>\n

Magnetic field inside a long straight current carrying conductor (at a point distant r from axis of the conductor)<\/strong><\/p>\n

Again, we choose a loop of radius r<\/em> as Amperian loop and choose direction of integration as anticlockwise.<\/p>\n

\"amperian-loop\"<\/p>\n

Magnetic Field Inside a Hollow Straight Current Carrying Conductor (at a Distance r from Axis of the Loop)<\/strong><\/p>\n

For the chosen Amperian loop (or radius r<\/em>), current enclosed is zero.<\/p>\n

\"amperian-loop-fig1\"<\/p>\n

Hence, there is no magnetic field inside a hollow current carrying conductor.<\/p>\n

Magnetic Field Due to An Infinite Plane Sheet of Current<\/strong><\/p>\n

\"plane-sheet-of-current\"<\/p>\n

An infinite sheet of current lies in x-z plane, carrying current along \u2013 z-axis. The field at any point P on y-axis is along a line parallel to x-z plane. We can take a rectangular Amperian loop as shown. If you traverse the loop in clockwise direction, inward current will be positive, by sign convention.<\/p>\n

\"ampere-circuital-law\"<\/p>\n

\n

Example : <\/strong><\/h4>\n

Consider a system of two long coaxial cylinders : Solid (of radius a carrying i<\/em> current in positive z<\/em> direction) and hollow (of radius b<\/em> carrying current i<\/em> in negative z direction).
\nFind the magnetic field at a point distant r from axis of the cylinders for<\/p>\n

    \n
  1. r \u2264 a<\/li>\n
  2. a < r < b<\/li>\n
  3. r \u2265 b<\/li>\n<\/ol>\n

    \"hollow\"<\/p>\n

    Solution : <\/strong><\/h4>\n

    <\/p>\n<\/div>\n

    Magnetic Field Inside a Long Solenoid<\/strong><\/p>\n

    A solenoid is a wire wound closely in the form of a helix, such that the adjacent turns are electrically insulated.<\/p>\n

    The magnetic field inside a very tightly wound long solenoid is uniform everywhere along the axis of the solenoid and is zero outside it.<\/p>\n

    \"magnetic-field-long-solenoid\"<\/p>\n

    Magnetic Field at a Point on The Axis of a Short Solenoid<\/strong><\/p>\n

    Consider a solenoid of length l<\/em> and radius R<\/em> containing N<\/em> closely spaced turns and carrying a steady current I<\/em>. We have to find out an expression for the magnetic field at an axial point P<\/em> lying in the space enclosed by the solenoid as shown in the figure below.<\/p>\n

    \"magnetic-field-short-solenoid\"<\/p>\n

    The field at a point on the axis of a solenoid can be obtained by the superposition of fields due to a large number of identical coils all having their centre on the axis of the solenoid.<\/p>\n

    Let us consider a coil of width dx<\/em> at a distance x from the point P<\/em> on the axis of the solenoid. The field at P<\/em> due to this coil is given by<\/p>\n

    \"magnetic-field-short-solenoid-a\"<\/p>\n

    If n<\/em> be the number of turns per unit length, dN = ndx<\/em>.<\/p>\n

    From the above figure,<\/p>\n

    \"magnetic-field-short-solenoid-b\"<\/p>\n

    Now consider some cases involving the application of above equation.<\/p>\n

    Case I : If the solenoid is of infinite length and the point is well inside the solenoid, \"solenoid-case-I\"<\/strong><\/p>\n

    Case II : If the solenoid is of infinite length and the point is near one end<\/strong><\/p>\n

    \"solenoid-case-II\"<\/p>\n

    Case III : If the solenoid is of finite length and the point is on the perpendicular bisector of its axis<\/strong><\/p>\n

    \"solenoid-case-III\"<\/p>\n

    \"solenoid\"<\/p>\n

    Magnetic Field of a Toroid<\/strong><\/p>\n

    A toroid is a solenoid bent to form a ring. Let N<\/em> be the total number of turns. For an Amperian loop of radius r,<\/em><\/p>\n

    \"magnetic-field-toroid\"<\/p>\n

    TAGS: <\/strong>Ampere’s Circuital Law<\/a><\/span>electric currents<\/a><\/span>electromagnetism<\/a><\/span>Magnetic Field of a Toroid<\/a><\/span>magnetic fields<\/a> <\/div>\n","protected":false},"excerpt":{"rendered":"

    This law is useful in finding the magnetic field due to currents under certain conditions of symmetry. Consider a closed  ….Read More >><\/a><\/p>\n","protected":false},"author":6,"featured_media":1235,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"om_disable_all_campaigns":false,"rank_math_lock_modified_date":false,"footnotes":""},"categories":[255,256,253],"tags":[369,371,372,373,370],"class_list":["post-988","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11th-physics","category-class-xiith","category-jee","tag-amperes-circuital-law","tag-electric-currents","tag-electromagnetism","tag-magnetic-field-of-a-toroid","tag-magnetic-fields"],"acf":[],"_links":{"self":[{"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/posts\/988","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/users\/6"}],"replies":[{"embeddable":true,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/comments?post=988"}],"version-history":[{"count":8,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/posts\/988\/revisions"}],"predecessor-version":[{"id":1075,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/posts\/988\/revisions\/1075"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/media\/1235"}],"wp:attachment":[{"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/media?parent=988"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/categories?post=988"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.meniit.com\/study-material\/wp-json\/wp\/v2\/tags?post=988"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}