Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Magnetostatics – Bar MagnetIron filings "map" of a bar magnet’s field Magnetic field in a bar magnetAs far back as 4500 years ago, the Chinese discovered that certain types of iron ore could attract each other and certain metals. Carefully suspended slivers of this metal were found to always point in the same direction, and as such could be used as compasses for navigation. The first compass is thought to have been used by the Chinese in about 376 B.C Greeks found this iron ore near Magnesia, in what is present day Turkey. It contained magnetite (Fe3O4), and came to be known as magnetic lodestone.Magnetostatics – Oersted’s Experiment Magnetic CompassA compass is an extremely simple device. A magnetic compass consists of a small, lightweight magnet balanced on a nearly frictionless pivot point. The magnet is generally called a needle. One end of the needle is often marked "N," for north, or colored in some way to indicate that it points toward north. On the surface, that's all there is to a compass In 1600, William Gilbert of England postulated that magnetic lodestones, or compasses, work because the earth is one big magnet. The magnetic field is generated by the spin of the molten inner core. The north end of a compass needle points to the geographic north pole, which corresponds to the earth’s south magnetic pole. The Earth can be thought of a gigantic bar magnet buried inside. In order for the north end of the compass to point toward the North Pole, you have to assume that the buried bar magnet has its south end at the North Pole. Magnetic CompassMagnetostatics – Oersted’s Experiment Magnetism and electricity were considered distinct phenomena until 1820 when Hans Christian Oersted conducted an experiment that showed a compass needle deflecting when in proximity to a current carrying wire. Oersted’s experiment In 1820, Hans Christian Oersted (1777-1851) used a compass to show that current produces magnetic fields that loop around the conductor It was observed that moving away from the source of current, the field grows weaker. Oersted’s discovery released flood of study that culminated in Maxwell’s equations in 1865.Magnetostatics – Biot-Savart’s Law Shortly following Oersted’s discovery that currents produce magnetic fields, Jean Baptiste Biot (1774-1862) and Felix Savart (1791-1841) arrived at a mathematical relation between the field and current. The Law of Biot-Savart is1 1 1221224I ddRp�=L aHNote: The Biot-Savart law is analogous to the Coulomb’s law equation for the electric field resulting from a differential charge 1 122212.4dQdRpe=aETo get the total field resulting from a current, you can sum the contributions from each segment by integrating 24.RIdRp�=�L aH(A/m)(A/m)Magnetostatics – An Infinite Line current Example 3.2: Consider an infinite length line along the z-axis conducting current I in the +az direction. We want to find the magnetic field everywhere.An infinite length line of current We first inspect the symmetry and see that the field will be independent of z and  and only dependent on ρ. So we consider a point a distance r from the line along the ρ axis. 24RIdRp�=�L aHThe Biot-Savart LawIdL is simply Idzaz, and the vector from the source to the test point is R zrr=- +R za a a( )( )32 22.4Idz zzrrp r�- ��- +=+�z za a aHThe Biot-Savart Law becomesMagnetostatics – An Infinite Line current Pulling the constants to the left of the integral and realizing that az x az = 0 and az x aρ = a, we have( )32 224Idzzfrpr�- �=+�aHThe integral can be evaluated using the formula given in Appendix D( )3 22 2 22 2dx xa x ax a=++�2 2 2 2 2 2 2 2 22t t tIt t tr r r r r r� � � �� � � �= + =� � � �+ + +� � � �( )32 2222 22 22 2 21 1dzzItrrr rr r�- �+= = = =� � � �+ +� � � ��� � � ��t  2222222222222322ttttzzzzzdzIttttttWhen the limitMagnetostatics – An Infinite Line current 2Ifpr=aHWe find the magnetic field intensity resulting from an infinite length line of current is An infinite length line of current ( )32 2222dzzIrr�- �+= =�( )32 224Idzzfrpr�- �=+�aHUsingMagnitude: The magnitude of the magnetic field is inversely proportional to radial distance.1 fr�H aDirection: The direction of the magnetic field can be found using the right hand rule.Magnetostatics – A Ring of CurrentExample 3.3: Let us now consider a ring of current with radius a lying in the x-y plane with a current I in the +az direction. The objective is to find an expression for the field at an arbitrary point a height h on the z-axis. ( )( )( )( )2 2z z3 32 2 2 22 20 044Iad h a d h aIah a h ap pf r f rffpp= =� - � -=+ +=� �a a a a a aH24RIdRp�=�L aHThe Biot-Savart LawRzh ar= -RR = a a aThe differential segment The vector drawn from the source to the test point is 2 2R ah= +( )Rzh ar= -Ra a aMagnitude:Unit Vector:dL = ada The biot-Savart Law can be written asMagnetostatics – A Ring of CurrentWe can further simplify this expression by considering the symmetry of the problem ( )( )( )( )2 2z3 32 2 2 22 20 04 4zd h a d h ah a h aIa Iap pf r rffp p= =� - ++ +=� �a a a a aH =A particular differential current element will give a field with an aρ component (from a x az) and an az component (from a x –aρ). Taking the field from a differential current element on the opposite side of the ring, it is apparent that the radial components cancel while the az components add. ( )22z32 2204Iadh apfp=+�aH( )2z32 222Iah a=+H az2Ia=H aaρ ComponentsCancelaz components addAt h = 0, the center of the loop, this equation reduces to HMagnetostatics – A SolenoidSolenoids are many turns of insulated wire coiled in the shape of a cylinder. A solenoid Suppose the solenoid has a length h, a radius a, and is made up of N turns of current carrying wire. For tight wrapping, we can consider the solenoid to be made up of N loops of current. To find the magnetic field intensity from a single loop at a point P along the axis of the solenoid, from we have ( )2z32 22d2 'dIaaz=+PH aThe differential amount of field resulting from a differential amount of current is given byThe differential amount of current can be considered a function