Physics 217, Fall 2002 27 November 2002(c) University of Rochester 127 November 2002 Physics 217, Fall 2002 1Today in Physics 217: energy in magnetic fieldsMagnetic loops in a solar active region, 9 May 1998, seen by the NASA TRACE satellite.150000 km27 November 2002 Physics 217, Fall 2002 2Energy in magnetic fieldsIn electrostatics,we found that the potential energy of an arrangement of charges, and the potentials and fields they create, was given byif the product of E, V, and r2approach zero as(see lecture notes for 25 September 2002). What have we for magnetism that corresponds to this?2allspace1in general, and218WVdEdρττπ==∫∫Vr →∞27 November 2002 Physics 217, Fall 2002 3Energy in magnetic fields (continued)To find such an expression, let’s exploitand consider a surface S, threaded by magnetic field and bounded by curve C:ThusBut so22,WLI=()1, so .BdddcLI LI dcΦ= ⋅ = × ⋅ = ⋅==⋅∫∫ ∫∫Ba A a AAvvSS CC— AA()211.22 2IWLI d dcc== ⋅= ⋅∫∫AAIAvvCCA,da′=∫IJvS()1.2Wdcτ=⋅∫JAV1cf. 2WVdρτ=∫VPhysics 217, Fall 2002 27 November 2002(c) University of Rochester 227 November 2002 Physics 217, Fall 2002 4Energy in magnetic fields (continued)We can use Ampère’s law to eliminate J from this expression,and simplify the result using Product Rule #6:to get1,24cWdcτπ=×⋅∫BAV—()()()() ()2,,B⋅×=⋅×−⋅×⋅×= −⋅×AB B A A BAB AB—————()()()221811,88WB dBd dτπτππ′′=−⋅×=−×⋅∫∫∫ABAB avVVS—27 November 2002 Physics 217, Fall 2002 5Energy in magnetic fields (continued)where we have used the divergence theorem, and identified the surface S” that bounds V. Now suppose we extend the volume Vto fill all space -- which we can do without penalty, if no additional currents are encompassed thereby. Then the surface integral (over S”) vanishes, since A and B approach zero very far from the currents. This is a cleaner case than that of electrostatics. Remember that this step runs into trouble with point electric charges (lecture notes, 25 September 2002), for which the surface integral doesn’t vanish. There aren’t any point magnetic charges. In fact, there aren’t any magnetic charges (monopoles) of any extent. 27 November 2002 Physics 217, Fall 2002 6Energy in magnetic fields (continued)SoIf there’s both E and B present, we thus get the neat result2allspace1.8WBdτπ=∫2allspace1cf. 8WEdτπ=∫()22allspace1.8WEBdτπ=+∫2200allspace1 in MKS units.2BWEdετµ=+∫Physics 217, Fall 2002 27 November 2002(c) University of Rochester 327 November 2002 Physics 217, Fall 2002 7Happy
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