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L29_a - AMPERE'S LAW(22.9 in text Ampere's law relates...

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AMPERE’S LAW (22.9 in text) Ampere’s law: relates magnetic field around a closed path to current through a surface bounded by that path Useful for calculating B r only in case of high symmetry o Similar tool for magnetic fields that Gauss’s law is for electric fields Plan: 1 st : state Ampere’s Law 2 nd : explain what it means 3 rd : justify by showing that it gives previous result for field around a long wire
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WHAT IS AMPERE’S LAW? loop through 0 loop closed around I s d B µ = r r WHAT DOES THIS MEAN? Look at a closed path (any shape) around a current I At each point around path, can take scalar product of B r with small segment of path s d r result: s d B r r loop closed s d B r r means sum s d B r r for all segments around the loop Ampere’s law says that resulting sum equals I 0 µ where I is the TOTAL current “threading” through the closed loop. I out of screen/page B B B ds ds ds
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AMPERE’S LAW: loop through 0 loop closed around I s d B µ = r r True for ANY loop (called Amperian loop) around a steady current BUT: Only helpful for calculating field for special (symmetric) cases o i.e. ones for which B can be factored out of the integral Cases where Ampere’s law might be useful: o B r is known constant over part of the path (by symmetry) o B r parallel to s d r so that ds B s d B = r r
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