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Magnetization Magnetization of a substance M is its magnetic moment per unit volume (similar to polarization in case of dielectrics in electric fields) M total V Total magnetic field at a point is a sum B B 0 0M All equations can be adapted by replacing 0 K m 0 Small magnetic effects are linear: m Km 1 0 for diamagnetics Magnetic susceptibility 0 for paramagnetics • Diamagnetism occurs in substances where magnetic moments inside atoms all cancel out, the net magnetic moment of the atom is zero. The induced magnetic moment is directed opposite to the applied field. Diamagnetism is weakly dependent on T. • Diamagnetic (induced atomic moment) effect is overcome in paramagnetic materials, whose atoms have uncompensated magnetic moments. These moments align with the applied field to enhance the latter. Temperature T wants to destroy alignment, hence a strong (1/T) dependence. B M=C Curie's Law T Magnetic effects are a completely quantum-mechanical phenomenon, although some classical physics arguments can be made. Example: Magnetic dipoles in a paramagnetic material Nitric oxide (NO) is a paramagnetic compound. Its molecules have maximum magnetic moment of ~ B . In a magnetic field B=1.5 Tesla, compare the interaction energy of the magnetic moments with the field to the average translational kinetic energy of the molecules at T=300 K. U max B B 1.4 1023 J 8.7 105 eV 3 K kT 6.2 1021 J 0.039 eV 2 Ferromagnetism • In ferromagnetic materials, in addition to atoms having uncompensated magnetic moments, these moments strongly interact between themselves. • Strongly nonlinear behavior with remnant magnetization left when the applied field is lifted. Permeability Km is much larger, ~1,000 to 100,000 Alignment of magnetic domains in applied field Hysteresis and Permanent Magnets Magnetization value depends on the “history” of applied magnetic field Example: A ferromagnetic material A permanent magnet is made of a ferromagnetic material with a M~10 6 A/m The magnet is in the shape of a cube of side 2 cm. Find magnetic dipole moment of a magnet. Estimate the magnetic field at a point 10 cm away on the axis total MV 8 A m 2 3 B ~ 0 total 10 T 10 G 3 2 x Magnetization curve for soft iron showing hysteresis Experiments leading to Faraday’s Law Electromagnetic Induction – Time-varying magnetic field creates electric field Changing Magnetic Flux No current in the electromagnet – B=0 - galvanometer shows no current. When magnet is turned on – momentarily current appears as B increases. When B reaches steady value – current disappears no matter how strong B field is. If we squeeze the coil as to change its area – current appears but only while we are deforming the coil. If we rotate the coil, current appears but only while we are rotating it. If we start displacing the coil out of the magnetic field – current appears while the coil is in motion. If we decrease/increase the number of loops in the coil – current appears during winding/unwinding of the turns. If we turn off the magnet – current appears while the magnetic field is being disappearing The faster we carry out all those changes - the greater the current is. Faraday’s Law quantified d B for a single - loop coil dt d B N for an N - loop coil dt B BA cos Anything changing magnetic flux will produce the effect Emf and Current Induced in a Loop d B d ( BA) dB A 0.24mV dt dt dt I R 0.048mA If the loop is made of the insulator, induced emf is still the same But the resistance is large, so little (or no current) is flowing Circuit with induced EMF only Area A1 with field B1 yield A1 B1 R1 A2 B2 R3 R2 I3 I1 I1 – I3 Kirchhoff’s rules still apply! It is only the origin of the EMFs that is different here from ordinary batteries. dB1 dt dB2 Likewise, 2 A2 dt Standard equations for loops : induced EMF 1 A1 I1 R1 I 3 R3 1 ( I1 I 3 ) R2 I 3 R3 2 And, e.g., it follows that R2 1 R1 2 I3 R1 R2 R1 R3 R2 R3 Direction of the induced EMF Alternating current (ac) generators B BA cos BA cos t BA sin t Direct current (dc) generators Split ring (commutator) does the job of reversing polarity every half cycle Motional emf – conductor moving in a constant magnetic field B Blx FB qvB will move charges until compensated by the electric field of end accumulations qvB qE qV /l V Bvl dx Bl Blv dt Generators as Energy Converters I Blv / R Presistor I 2 R ( Blv )2 / R Generator does not produce electric energy out of nowhere – it is supplied by whatever entity that keeps the rod moving. All it does is to convert it to a different form, namely to electric energy (current) Who does the work? We! - By moving the bar: Papplied Fv IBlv ( Blv )2 / R Energy conserved After initial push, Rotating bar : velocity w ill relax v(r ) r Small element : decelerate d by the magnetic force : Bv dr d Total emf : l Bl 2 Br dr 2 0 2 m dv ( Bl ) IBl v dt R v v0 exp( t / ) mR /( Bl ) 2 Motion does not necessarily mean changing magnetic flux! Significance of the minus sign – Lenz’s Law Induced current has such direction that its own flux opposes the change of the external magnetic flux Magnetic field of the induced current wants to decrease the total flux Magnetic field of the induced current wants to increase the total flux Correspondingly, magnetic forces oppose the motion – consistently with conservation of energy! Lenz’s Law – the direction of any magnetic induction effect as to oppose the cause of the effect Lenz’s Law – a direct consequence of the energy conservation principle Finding the direction of the induced current Induced Electric Fields No matter wha t , the total force on a charge is F q (E v B ) To have current in the loop, F 0 We did explain currents in moving conductors (" motional emf" ) with FB qv B BUT! Faraday' s experiment s show that currents are induced when v 0 but B B(t ) What is it that drives charges then? Electric field E induced by changing B ! emf is nothing but the work done to move a unit charge around the loop once, which is the line integral around the loop E ds