{"id":5261,"date":"2024-06-29T00:16:10","date_gmt":"2024-06-29T00:16:10","guid":{"rendered":"https:\/\/vartmaaninstitutesirsa.com\/?page_id=5261"},"modified":"2024-06-29T00:19:36","modified_gmt":"2024-06-29T00:19:36","slug":"amperes-circuital-law","status":"publish","type":"page","link":"https:\/\/vartmaaninstitutesirsa.com\/index.php\/amperes-circuital-law\/","title":{"rendered":"Ampere\u2019s circuital law"},"content":{"rendered":"
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14 Ampere\u2019s circuital law:-<\/u><\/strong>\u00a0m.imp<\/sup><\/u><\/strong>\u00a0<\/u><\/strong><\/p>\n

According to Ampere\u2019s circuital law, the line integral of magnetic field around a closed circuit is equal to\u00a0\"\"\u00a0times the total current flowing through the circuit.<\/p>\n

I,e.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\"\"\u00a0\u00a0\u00a0=\"\"I<\/p>\n

Proof: \u2013<\/u>\u00a0\u00a0\u00a0\u00a0<\/u><\/p>\n

\"F:\\unitSuppose a Ampere\u2019s surface around a straight conductor carrying current I. Now the magnet field at P due to current I in conductor can be given by<\/p>\n

\"\"<\/p>\n

Now the line integral of\"\"\u00a0is<\/p>\n

\"\"<\/p>\n

=\u00a0\"\"\u00a0\"\"<\/p>\n

Here\u00a0\u00a0\u00a0\"\"\u00a0=\u00a0\"\"= circumference of the circle<\/p>\n

So\u00a0\"\"<\/p>\n

=\u00a0\"\"\u00a0\"\"<\/p>\n

Or\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\"\"\u00a0=\u00a0\"\"<\/p>\n

This is required expression for Ampere circuital law.<\/p>\n

\uf0fc\u00a0\u00a0\u00a0\u00a0\u00a0Ampere\u2019s circular law holds for DC or steady current which does not change with time.<\/em><\/p>\n

\uf0fc\u00a0\u00a0\u00a0\u00a0\u00a0Ampere\u2019s circuital law for magnetic field is analogy to Gauss law in electrostatics.<\/em><\/p>\n

\uf0fc\u00a0\u00a0\u00a0\u00a0\u00a0Biot Savart\u2019s law & Ampere\u2019s circuital law are equivalent in same sense as Coulombs Law & Gauss theorem in electrostatics.<\/em><\/p>\n

\uf0fc\u00a0\u00a0\u00a0\u00a0\u00a0For a loop that is not connected to the current<\/em>\u00a0\u00a0\u00a0<\/em>\"\"<\/p>\n

 <\/p>\n

15 Applications of Ampere\u2019s Circuital law<\/u><\/strong>\u00a0m.imp<\/sup><\/u><\/strong>\u00a0<\/u><\/strong><\/p>\n

1. Magnetic field due to an infinite long current carrying wire:-<\/u><\/p>\n

\"F:\\unitSuppose an infinite long straight conductor carrying current I. Now we have to calculate magnetic field at P on the Amperean loop of radius r. Consider small length\u00a0\"\"\u00a0at loop, then magnetic field B at\u00a0\"\"\u00a0can be given by Ampere\u2019s circuital law as<\/p>\n

\"\"\u00a0\u00a0\u00a0=\u00a0\"\"<\/p>\n

O r\u00a0\u00a0\"\"cos00<\/sup>\u00a0\u00a0\u00a0=\"\"<\/p>\n

Or\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0B\u00a0\"\"=\"\"<\/p>\n

Here\u00a0\"\"\u00a0= circumference of circle =\u00a0\"\"<\/p>\n

\uf0f0\u00a0\u00a0\u00a0\u00a0B\u00a0\"\"\u00a0\"\"\u00a0\u00a0\u00a0=\u00a0\u00a0\"\"<\/p>\n

\uf0f0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0B<\/strong>\u00a0\u00a0\u00a0<\/strong>=<\/strong>\u00a0\u00a0\u00a0<\/strong>\"\"<\/p>\n

 <\/p>\n

\"\"(b) Solenoid: \u2013<\/u><\/strong>\u00a0m.imp<\/sup><\/u><\/strong><\/p>\n

Solenoid means an insulated copper wire wounded closely in the form of a helix. The length of the solenoid is very large as compared to its diameter<\/em>.<\/p>\n

Calculation of magnetic field inside a long straight solenoid:-<\/u><\/strong><\/p>\n

\"F:\\unitSuppose a rectangular Amperean loop\u00a0\"\"\u00a0of length\"\". If N is the number of turns in the length\"\", then total current through the loop equal to\u00a0\"\".<\/p>\n

Thus according to Amperes law\u00a0\u00a0\u00a0<\/strong><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/strong>\"\"\u00a0\u00a0=\u00a0\"\"\u00a0\u2013(1)<\/p>\n

\"RightOr\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u0283abcda<\/sub>\u00a0\"\"\u00a0\u00a0=\u00a0\u00a0\u0283a<\/sub>b<\/sup>\u00a0\"\"+\u00a0\u00a0\u0283b<\/sub>c<\/sup>\u00a0\"\"+\u00a0\u00a0\u0283c<\/sub>d<\/sup>\u00a0\"\"+ \u0283d<\/sub>a<\/sup>\u00a0\"\"<\/p>\n

Here\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u0283b<\/sub>c<\/sup>\u00a0\"\"= \u0283d<\/sub>a<\/sup>\u00a0\"\"=\u00a0\u00a0\u0283\u00a0\"\"cos\"\"\u00a0\u00a0= 0<\/p>\n

Also\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u0283c<\/sub>D<\/sup>\u00a0\"\"\u00a0\u00a0= 0 because CD lies outside the solenoid.<\/p>\n

\"\"\u00a0= \u0283a<\/sub>b<\/sup>\u00a0\"\"\u00a0= B\"\"\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0..(2)<\/p>\n

Comparing eqn<\/sup>\u00a01 & 2 we get<\/p>\n

B\"\"\u00a0=\u00a0\"\"<\/p>\n

Or\u00a0\u00a0\u00a0B =\u00a0\"\"<\/p>\n

Or\u00a0\u00a0\u00a0\u00a0B =\u00a0<\/strong>\"\"<\/p>\n

Where n is the number of turns per unit length.<\/p>\n

Q<\/strong>8.<\/strong>How is the magnetic field inside a given solenoid made strong?<\/strong><\/p>\n