Fields, Units, Magnetostatics European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Institut N´ eel CNRS-Universit´ e Grenoble Alpes 10 octobre 2017 European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Motivation Magnetism is around us and magnetic materials are widely used Magnet Attraction (coins, fridge) Contactless Force (hand) Repulsive Force : Levitation Magnetic Energy - Mechanical Energy (Magnetic Gun) Magnetic Energy - Electrical Energy (Induction) Magnetic Liquids A device full of magnetic materials : the Hard Disk drive European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
reminders Flat Disk Rotary Motor Write Head Voice Coil Linear Motor Read Head Discrete Components : Transformer Filter Inductor European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics How to describe Magnetic Matter ? How Magnetic Materials impact field maps, forces ? How to model them ? Here macroscopic, continous model Next lectures : Atomic magnetism, microscopic details (exchange mechanisms, spin-orbit, crystal field ...) European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics w/o magnets : Reminder Up to 1820, magnetism and electricity were two subjects not experimentally connected H.C. Oersted experiment (1820 - Copenhagen) European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics induction field B Looking for a mathematical expression Fields and forces created by an electrical circuit (C1, I) Elementary � dB induction field created at M dB ( M ) = µ 0 I � Biot and Savart law (1820) � dl ∧ � u 4 π r 2 European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Vocabulary dB ( M ) = µ 0 I � dl ∧ � u � 4 π r 2 � B is the magnetic induction field � B is a long-range vector field ( 1 1 r 2 becomes r 3 for a closed circuit). European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Force Force created by (C1, I) on (C2, I’) dB dl’ u r dl M I I’ (C1) (C2) dF ( M ) = I ′ � dl ′ ∧ � Laplace Law � B ( M ) What is the Force between 2 parallel wires carrying the same current I : attractive/repulsive ? definition for Amp` ere : 1 A if 2 parallel wires 1m apart and force is f=2 10 − 7 N/m. European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Motor Origin of the electric-mechanical transducer = motors (linear and rotary motors) Synchronous Motor (dc current rotor, ac current stator). Downsizing, Mechanical Torque, Energy Yield, Move to permanent magnet rotors. European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : units dF ( M ) = I ′ � � dl ′ ∧ � B ( M ) Using SI units : Force F Newton(N) Intensity Amp` ere (A) Magnetic Induction B Tesla (T) so 1 T = 1 NA − 1 m − 1 and µ o = 4 π 10 − 7 NA − 2 exact value European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetic Induction � B Some magnetic induction � B properties n dS (V) (S) �� B · � � dS = 0 S � B flux is conservative B lines never stop (closed B loops) ! B flux is conserved. It is a relevant quantity with a name : Wb(Weber) = T.m 2 (B-field is sometimes called the magnetic flux density) European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : � B � B flux conservation is equivalent to one of the local Maxwell equation : ∇ · � � B = 0 B can be derived from a vector potential � � A so that � ∇ × � B = � A For the preceding circuit : � A = µ 0 I � dl � 4 π r ( C 1) applying the curl operator one comes back to � B Note : � A is not unique. � � A ( � r ) + grad φ ( � r ) is also solution A gauge can be chosen (i.e. � ∇ · � A = 0, Coulomb gauge) European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : � A This is equivalent to the role of the electric potential V in electrostatics with � E = − � grad V (numerical simulation interest) European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : B is an pseudo-vector Mirror symmetry for a current loop : � B is a axial vector. � B is NOT time-reversal invariant, unlike electrostatics. European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Ampere ’s theorem j ( ) � dS (S) dl Ampere Theorem � B · � � dl = µ 0 I no magnet (Γ) (Γ) � H · � � Note : with magnetic materials it becomes : dl = I European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Amp` ere theorem Similar to � B flux conservation Amp` ere theorem has a local equivalent (Maxwell) ∇ × � � B = µ 0 � j j is the volume current density (A/m 2 !) where � European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Application Ampere theorem Application to the infinite straight wire � B · � � dl = µ 0 I (Γ) B = µ 0 I � 2 π r � u θ European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : magnetic moment Current Carrying Loop Magnetic Moment M r n S I Circular Loop (radius R), carrying current I, oriented surface � S m = � S · I unit A.m 2 Its magnetic moment is � European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Dipolar Approximation M r n S I When r >> R , � B created by the loop becomes µ 0 � B = 4 π r 3 (2 mcos θ� u r + msin θ� u θ ) European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Dipolar Approximation µ 0 � B = 4 π r 3 (2 mcos θ� u r + msin θ� u θ ) can also be written along � r and � m : B = µ 0 4 π (3( � m · � r ) � r − � m � r 3 ) r 5 Earth Field = Dipolar Field (good approximation). European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Earth Field Geographic North Pole is Magnetic South Pole online model : www.ngdc.noaa.gov European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Earth Field The magnetic pole moves up toward Russia. Presently (86 ◦ N, 159 ◦ W), its speed is 55 km/year to N-NW. European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Electrostatics Analogy The magnetic dipolar field is equivalent to the electric dipolar field p = q � One defines an electric dipole � l and 1 � 4 πǫ 0 r 3 (2 pcos θ� u r + psin θ� E = u θ ) For an elementary loop � m is the loop magnetic dipole . European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Field lines E B + - + Fields around an electric dipole and a magnetic dipole European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Reciprocity Theorem How to optimise the signal sensed by a coil close to the sample ? m = I 2 . � � S 2 Signal = flux of induction created by sample � m through C 1 φ 21 = � B 2 (1) .� S 1 Mutual inductance M 12 equals M 21 European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Reciprocity Theorem φ 21 = � B 2 (1) .� S 1 φ 21 = M . I 2 et φ 12 = M . I 1 φ 21 = φ 12 . I 2 / I 1 = � B 1 (2) .� S 2 . I 2 / I 1 = � B 1 (2) . � m / I 1 The sample � m creates a B-flux in the detection coil equal to the m and � scalar product � B at � m assuming 1 A in the detection coil. European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics with Magnets : Magnetisation Experimental Facts : So-called magnetic materials produce effects similar to the ones created by electric circuits. Iron filings + magnet equivalent to Iron filing (or compass) and solenoid European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
Magnetostatics : Magnetisation A magnetic material will be modeled as a set of magnetic dipoles. � ∆ � m = m i � i Magnetisation � M is the magnetic moment per unit volume : M = ∆ � m � ∆ V Average over 1 nm to smoothen the atomic contributions (continuous model). m = I · � unit is A · m 2 Magnetic Moment � S M = ∆ m unit is A · m − 1 Magnetisation ∆ V European School on Magnetism Laurent Ranno (laurent.ranno@neel.cnrs.fr) Fields, Units, Magnetostatics
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