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Phys2212_32.8+to+32.10 - Physics 2212 Electricity and...

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Physics 2212 Electricity and Magnetism Lecture 20 (Knight: 32.8-.10) More Magnetic Effects
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10/06/09 Physics 2212 - Lecture 120 Line Integrals Made Easy If B is everywhere perpendicular to the path of integration ds , then: 0 f i B ds = r r If B is everywhere parallel to thepath of integration ds , then: f i B ds BL = r r
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10/06/09 Physics 2212 - Lecture 120 Ampere’s Law A special caseof a lineintegral is onethat runs in a closed path and returns to whereit started, i.e., a lineintegral around a closed curve, which, for a magnetic field, is denoted by: Consider thecaseof thefield at a distance d from a long straight wire: 0 2 4 I B d μ π = This result is: independent of theshapeof thecurvearound thewire; independent of wherethecurrent passes through the curve; depends only on theamount of current passing through theintegration path. Ampere’s Law Bds r r O Bds r r O = 2 π r B = μ 0 I Bds r r O = μ 0 I
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10/06/09 Physics 2212 - Lecture 120 Example : The Magnetic Field Inside a Current-Carrying Wire A wireof radius R carries current I uniformly distributed across its cross section. Find themagnetic field insidethewireat a distancer<R from theaxis. 2 2 through 2 2 I r I JA r I R R π π = = = 2 0 through 0 2 (2 ) B ds B ds B r r I I R π μ μ = = = = r r Ñ Ñ 2 0 0 2 2 2 2 4 I r Ir B r R R μ μ π π = = Bds r r O
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10/06/09 Physics 2212 - Lecture 120 Symmetry and Long Solenoids Original Solenoid Rotate180 0 about y axis Reverse Current Therefore, radial B field components near center areruled out by symmetry. But wecan still haveB fields in z and θ directions. Radial B field? y Can B have a radial component inside solenoid ?
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10/06/09 Physics 2212 - Lecture 120 The Magnetic Field of a Solenoid (1) A solenoid is a helical coil of wire consisting of multipleloops, all carrying thesamecurrent. Onecan think of thefield of a solenoid by superimposing thefields from several loops, as shown in thelower figure. On theaxis, thethreefields will add to makea stronger net field, but outsidetheloop thefields from loops 1 and 3 will tend to cancel thefield from coil 2. When thefields from all theloops aresuperimposed, theresult is that thefield insidethesolenoid is strong and roughly parallel to the axis, whilethefield outsideis very weak. In thelimit of an ideal solenoid thefield insideis uniform and parallel to theaxis, whilethe field outsideis zero.
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10/06/09 Physics 2212 - Lecture 120 The Magnetic Field of a Solenoid (2) Wecan useAmpere’s Law to calculatethefield of an ideal long solenoid by choosing theintegration path carefully. Wechoosea rectangular LxW loop, with onehorizontal sideoutsidethe solenoid and thevertical sides passing through.
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