Physics 2102 Jonathan Dowling Lecture 17 THU 18 MAR 10 Ampere s law QuickTime and a TIFF LZW decompressor are needed to see this picture Andr Marie Amp re 1775 1836 Remember Gauss Law for EFields Given an arbitrary closed surface the electric flux through it is proportional to the charge enclosed by the surface q q Flux 0 r r q E dA 0 Surface Gauss s Law for B Fields No isolated magnetic poles The magnetic flux through any closed Gaussian surface will be ZERO This is one of the four Maxwell s equations B d A 0 There are no SINKS or SOURCES of B Fields What Goes IN Must Come OUT Ampere s law Closed Loops r r B ds 0ienclosed LOOP The circulation of B the integral of B scalar ds along an imaginary closed loop is proportional to the net amount of current traversing the loop i4 i2 i3 i1 ds r r B d s 0 i1 i2 i3 loop Thumb rule for sign ignore i4 If you have a lot of symmetry knowing the circulation of B allows you to know B Ampere s law Closed Loops r r B ds 0ienclosed LOOP The circulation of B the integral of B scalar ds along an imaginary closed loop is proportional to the net amount of current traversing the loop r r B d s 0 i1 i2 loop Thumb rule for sign ignore i3 If you have a lot of symmetry knowing the circulation of B allows you to know B Calculation of ienc We curl the fingers of the right hand in the direction in which the Amperian loop was traversed We note the direction of the thumb All currents inside the loop parallel to the thumb are counted as positive All currents inside the loop antiparallel to the thumb are counted as negative All currents outside the loop are not counted In this example ienc i1 i2 Sample Problem Two square conducting loops carry currents of 5 0 and 3 0 A as shown in Fig 30 60 What s the value of B ds through each of the paths shown Path 1 B ds 0 5 0A 3 0A Path 2 B ds 5 0A 5 0A 3 0A Ampere s Law Example 1 Infinitely long straight wire with current i R Symmetry magnetic field consists of circular loops centered around wire r r B ds 0i So choose a circular loop C so B is tangential to the loop everywhere Angle between B and ds is 0 Go around loop in same direction as B field lines C Bds B 2 R i 0 C Much Easier Way to Get B Field Around A Wire No Calculus 0i B 2 R Ampere s Law Example 2 Infinitely long cylindrical wire of finite radius R carries a total current i with uniform current density Compute the magnetic field at a distance r from cylinder axis for r a inside the wire r a outside the wire Current out of page circular field lines r r B ds 0i C i Ampere s Law Example 2 cont r r B d s i 0 C B 2 r 0ienclosed 0ienclosed B 2 r ienclosed Current out of page field tangent to the closed amperian loop Need Current Density J 2 i r 2 J r 2 r i 2 R R 0ir B 2 2 R 2 For r R For r R ienc i so B 0i 2 R LONG WIRE Ampere s Law Example 2 cont B 0i 2 R O R r r R r R B r B 1 r r R 0ir B 2 2 R r R 0i B 2 r Outside Long Wire Solenoids r r B ds 0ienc r r B ds 0 B h 0 0 ienc iN h i N L h inh r r B ds 0ienc Bh 0inh B 0in The n N L is turns per unit length are needed to see this picture Magnetic Field in a Toroid Doughnut Solenoid 0 Ni B 2 r QuickTime and a decompressor are needed to see this picture Toriod Fusion Reactor Power NYC For a Day on a Glass of H2O Magnetic Field of a Magnetic Dipole A circular loop or a coil currying electrical current is a magnetic dipole with magnetic dipole moment of magnitude NiA Since the coil curries a current it produces a magnetic field that can be calculated using Biot Savart s law r r 0 0 B z 2 2 3 2 2 R z 2 r z3 All loops in the figure have radius r or 2r Which of these arrangements produce the largest magnetic field at the point indicated Magnetic Dipole in a B Field QuickTime and a decompressor are needed to see this picture QuickTime and a decompressor are needed to see this picture r F 0 r r r B r r U gB Force is zero in homogenous field Torque is maximum when 90 Torque is minimum when 0 or 180 Energy is maximum when 180 Energy is minimium when 0 QuickTime and a Animation decompressor are needed to see this picture Neutron Star a Large Magnetic Dipole
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