Cose no. 4 The Theory of Electromagnetic Field Technical University of Cluj-Napoca http://www.et.utcluj.ro/cs_electromagnetics2006_ac.htm http://www.et.utcluj.ro/~lcret March 19-2009
Chapter 3 Magnetostatics fields
Introduction notes Magnetostatics fields
Magnetostatics fields Magnetic Fields (B) I Crents Permanent Magnets
Magnetostatics fields The origin of magnetism lies in moving electric charges. Moving (or rotating) charges generate magnetic fields. An electric crent generates a magnetic field. A magnetic field will exert a force on a moving charge. A magnetic field will exert a force on a conductor that carries an electric crent.
Magnetostatics fields Stationary charge: v q = 0 A stationary charge produces an electric field only. E 0 B = 0 Moving charge: v q 0 and v q = constant E 0 B 0 Accelerating charge: v q 0 and a q 0 A uniformly moving charge produces an electric and magnetic field. A accelerating charge produces an electric and magnetic field and a radiating electromagnetic field. E 0 B 0 Radiating field
Magnetostatics fields Units and definitions: 1 T = 10 4 G B v Magnetic field vector Magnetic induction Magnetic flux density Tesla Gauss H v Magnetic field strength SI unit Wb 1T = 1 2 m Weber v B = µ v H
Magnetostatics fields Permeability µ = µ µ r o Permeability of free space Relative permeability for a medium 7 H µ o = 4 π 10 m Exact constant Permeability of the medium H m Wb m
Magnetostatics fields Relative permeability µ r and susceptibility χ m Bismuth Mercy Gold Silver Lead Copper Water 0.99983 0.999968 0.999964 0.99998 0.999983 0.999991 0.999991-1.66 E-4-3.2 E-5-3.6 E-5-2.60 E-5-1.7 E-5-0.98 E-5-0.88 E-5 Diamagnetic Vacuum Air Aluminium Palladium Cobalt Nickel Iron 1.000 1.00000036 1.000021 1.00082 250 600 6000 0 3.6 E-7 2.5 E-5 8.2 E-4 --- --- --- Paramagnetic Ferromagnetic http://www.ndt-ed.org/educationresoces/communitycollege/magparticle/physics/magneticmatls.htm
Magnetostatics fields Crent loop Coil or solenoid
Magnetic flux Magnetostatics fields The magnetic flux through an open sface S is given by: Ψ = B d A, Weber = Wb S [ ] In an electrostatic field, the flux passing thought a closed sface is the same as the charge enclosed. Thus it is possible to have isolated electric charges, which also reveals that electric flux lines are not necessarily closed. Unlike electric flux lines, magnetic flux lines always close upon themselves. This is due to the fact that it is not possible to have isolated magnetic poles (or magnetic charges). Thus, the total flux through a closed sface in a magnetic field must be zero: B d A = 0 Σ This equation is referred to as the law of conservation of magnetic flux or Gauss s law for magnetostatic fields.
Definition: Magnetostatics fields Laws of magnetostatics The magnetic flux law The total magnetic flux through a closed sface is always zero. B d A = 0 Integral form of the law Σ Σ B d A = divb dv = 0 V Σ divb = 0 Differential form of the law u u div B = n12 B2 B1 = B2 B1 = 0 Boundary conditions ( ) s n n
Magnetostatics fields Laws of magnetostatics The magnetization law From experimental studies, it is found that the magnetization vector (the temporary component) is strongly related to the magnetic field strength. For most common magnetic materials, these two vectors are collinear and proportional for a wide range of values of H (linear materials and isotropic). χ m u u M t = χ H Valid just for linear materials where: is the magnetic susceptibility of the material. m When the magnetic susceptibility depends on the magnetic field strength H, it is said that the medium is nonlinear, because all the field relations become nonlinear equations. When the magnetic susceptibility depends on the position in the volume of the magnetic body, it is said that the problem is inhomogeneous, as opposed to the homogeneous case when the properties of the material are constant throughout the volume. Moreover, the magnetic properties may depend on the direction of the applied field. This is called anisotropy of the magnetic material.
Magnetostatics fields Laws of magnetostatics The relation between B, H and M vectors The vector sum, between the magnetization vectors (both components) and the magnetic field strength, multiplied with the permeability of the vacuum, is equal, at any moment and point, with the magnetic flux density: u uu uu B= H + M + M µ 0 ( ) t p u uu For materials without permanent magnetization: B= µ ( ) 0 H + M t For linear materials without permanent magnetization: u uu u u u u B= µ H + M = µ H + χ H = µ + χ H = µ H ( ) ( ) t m ( m) 0 0 0 1 For materials with anisotropy and without permanent magnetization: u B = µ H
Magnetostatics fields Laws of magnetostatics The relation between B, H and M vectors Then, the relation between the magnetic flux vector and the magnetic field strength is a tensor one: B = µ u H B µ µ µ x xx xy xz H x = µ µ µ By yx yy yz H y B µ µ µ z zx zy zz H z Fortunately, it often suffices to assume that the medium is homogeneous, linear and isotropic. This is the simplest possible case. Final note on the physical meaning of the relative magnetic permeability: it shows how many times the magnetic field strength is decreased or increased in the volume of the magnetic material due to the effect of the magnetization. Note: In general the magnetization vector consists in 2 components: - a temporary component (M t ) and a permanent one (M p )
Magnetostatics fields C 1 C 2 i r r s ds dθ e θ Ampere s law H H Determine the circulation of H - field along an arbitrary circuit c u r u r H d s = H d s = i C1 C 2 =r. dθ Consequences: 1. More crents through C 2 add up; 2. Crents outside C 2 do not contribute; 3. Position of crent inside C 2 is not important.
Magnetostatics fields Ampere s law can be derived in its differential form as follows: u r u r H d s = i H d s = J d A C C S S C 1 C u u roth d A = J d A roth = J S Ampere s differential form (v = 0) C It is now clear that the electric crents are the cl-soces of the magnetic filed. Notice that the magnetic field has a non-zero cl, and this non-zero cl equals the electric crent density, unlike the electrostatic field, which is a cl-free field. u roth = J rote = 0
Magnetostatics fields Irrotational Field Rotational Field Electric field rote = 0 Magnetic field u roth = J
Magnetostatics fields Note: until now we have considered that the sface S is immobile. In the most general case of media in movement (sface has a relative speed towards the media), the Ampere's law in differential form must be completed as: u D r r roth = J + + ρv v+ rot D v A m t ( ) 2, / Conduction crent density Displacement crent density Convection crent density Roentgen crent density From the physical point of view, the correction crent appears due to the displacement, with the speed v (towards the sface S C ), of bodies charged with charges (having the volume density of the charge ρ v ) and the Roentgen crent appears due to the displacement with the speed v of polarized bodies (having volumetric density of the polarized charge ).
C( S) Magnetostatics fields u D r r roth = J + + ρv v + rot D v t ( ) u r D r r H ds = J + + ρv v+ rot( D v) da t S u H ds r = i+ id + ic + ir C( S) In the most general case, a magnetic field can be produced by: conduction crents displacement crents convection crents Roentgen crents Ampere's integral form
Flux linkage Magnetostatics fields
Magnetostatics fields Flux linkage The flux linkage may be self flux linkage and mutual flux linkage. A single coil has only its own self flux linkage, i.e. this is the flux created by its own crent, which flows through its own tns. The mutual flux linkage is defined only if a pair of magnetically coupled coils exists. The mutual flux linkage of coil 1 is due to the magnetic field of the coil 2, which induced emf in coil 1.
Magnetostatics fields
Magnetostatics fields Ψ 11 = u B1 d A1, Wb S 1 u µ I ds r r B = 4 π r 1 1 1, 3 C 1 T r µ I d s r Ψ = da Wb Ψ I 4 π r S1 C1 1 1 11, 3 11 µ 1
Magnetostatics fields L u B da H da, Ψ SC [ ] SC [ ] = = u r = µ u r I H ds H ds C C
[ ] u H da Magnetostatics fields [ ] SC L = µ u r, H H ds C E da C = ε Σ 2 r, F E ds 1 [ ] Just as it was shown that for capacitors: It can be shown that: Lext C = µ ε C G ε = σ L ext Note: represents only the external inductance, related to the external flux of an inductor. In an inductor such as a coaxial or parallel-wire transmission line, the inductance produced by the flux internal to the conductor is called the internal inductance L int while that produced by the flux external to it is called external inductance. The total inductance L is: L= L + L int ext
Magnetostatics fields Magnetostatic field energy W n 1 = ψ i 2 m i i The above relation gives us the expression of the energy stored in the magnetic field of several electric circuits due to crents i= 1 W 1 1 1 = = µ = µ = V 2 2 2 e 2 wm H H H B H Magnetic field density In, general: wm 1 u = B H 2 1 u Wm = we dv = B H dv 2 V V
Magnetostatics fields Analogy Electrostatics Magnetostatics Scalar Potential - V Vector Potential - A E-field Permitivity Volume Charge Density Sface Charge Density Capacitance - C Laplace s Equation Poisson s Equation H-field Permeability Crent Density Sface Crent Density Inductance - L Laplace s Equation Poisson s Equation
Electromagnetic field Introduction So far we have: dive = ρ εv rote = 0 divb = 0 u roth = J Electrostatic field Magnetostatic field These equations are OK for static fields, i.e. those fields independent of time. When fields vary as a function of time the cl equations acquire an additional term. rote = 0 gets a v B t u roth = J gets a D t
Electromagnetic field Faraday s Experiments N S i v Michael Faraday discovered induction in 1831. Moving the magnet induces a crent i. Reversing the direction reverses the crent. Moving the loop induces a crent. The induced crent is set up by an induced EMF.
Electromagnetic field Faraday s Experiments (left) i v = 0 di/dt (right) EMF Changing the crent in the right-hand coil induces a crent in the left-hand coil. The induced crent does not depend on the size of the crent in the right-hand coil. The induced crent depends on di/dt.
Electromagnetic field Faraday s Experiments 1) 2) N S i v Moving the magnet changes the flux Ψ (1) motional EMF. Changing the crent changes the flux Ψ (2) - transformer EMF. Faraday: changing the flux induces an EMF (e). e = dψ dt (left) Faraday s law i v = 0 (right) di/dt EMF The emf induced around a loop equals the rate of change of the flux through that loop
Electromagnetic field Faraday formulated the law named after his name The induced electromagnetic force (EMF) - e emf or simply (e), in any closed conducting loop (circuit) is equal to the time rate of change of the magnetic flux linkage of the loop. dψ d e= = B da dt dt SΓ Integral form of Faraday s law The negative sign shows that the induced emf (and crents) would act in such a way that they would oppose the change of the flux creating it. This law is also known as Lentz law of EMF induction. If the circuit consists of N loops all of the same area and if Ψ is the flux through one loop, then the total induced emf is: dλ dψ e= = N dt dt
Electromagnetic field Lenz s Law: The direction of any magnetic induction effect (induced crent) is such as to oppose the cause producing it. (Opposing change = inertia!) B (increasing) E A A B (decreasing) E A A E E B (increasing) B (decreasing)
Electromagnetic field Differential form of Faraday s law e = dψ dt e = E ds Ψ = B da Γ SΓ Note: S Γ is an open sface. Γ r d E ds = B da dt SΓ
Electromagnetic field Applying Stokes s theorem: Γ Γ r d E ds = B da dt SΓ r d E ds = rote da = B da dt SΓ SΓ Suppose that the sface S Г is mobile with the velocity v, then the derivative with respect the time of the sface integral will be: d dt B r B da = da + rot( B v) da t SΓ SΓ SΓ The sface is immobile The sface is mobile with velocity v
Electromagnetic field Γ r d E ds= B da dt SΓ d dt B r B da = da + rot( B v) da t SΓ SΓ SΓ B r rote da = da rot( B v) da t SΓ SΓ SΓ B r rot E = + rot v B t ( ) Differential form of Faraday s law
Electromagnetic field The induced electromagnetic force (emf) around a circuit can be separated into two terms: Transformer emf component, due to the time-rate of change of the B- field : r B B e rote = transformer = E ds= rote da= da t t Motional emf component, due to the motion of the circuit: r r r r ( ) E = v B emotional E ds v B ds Note: the induced electric field is non-conservative!!!! rote 0 Γ = = Γ The electric field is conservative only in electrostatics regime!!!! rote = 0 Γ S Γ S Γ
Particular cases: 1) The sface is immobile (v = 0): The motional emf is zero: Electromagnetic field Γ A E = gradv t r r r e E ds v B ds motional The transformer emf is non-zero: Differential form of Faraday s law: Γ ( ) 0 = = = B r A r e = = = transformer da E ds ds t t B rote = t SΓ Γ Γ 2) The sface is mobile but the magnetic field B is constant in time: r E = v B gradv The motional emf is non-zero: e E ds r v r B ds r motional Γ Γ ( ) = =
Electromagnetic field Maxwell s equations (v = 0) Γ Γ Integral form Differential form Significance E ds r Ψ = t u r D H ds = i+ da t Σ S B rote = t roth u = J + D t D da= Q divd = ρv B da = Σ 0 divb = 0 Faraday s law Maxwell 1 Ampere s law Maxwell 2 Electric flux s law Maxwell 3 Magnetic flux s law Maxwell 4
Electromagnetic field Maxwell s equations B rote = t u D roth =+ t = µ u H rote t =+ ε u E roth t
Electromagnetic field u H µ ( ) u rote = rot rote = µ roth t t u roth = + ε E t 2 2 ( ) E E rot rote = µ ε grad( dive) E = µ ε 2 2 t t In free space: ρv dive = = 0 ε 2 µ ε 2 E or E = µ ε u u H with: H = 2 2 t t 2 1 E E = 2 2 In vacuum: c t u Wave s equations u 2 1 H H = c 2 t 2 1 µ ε = µ ε 2 r r c
u roth u roth =+ ε E t Electromagnetic field E t = µ u H rote t rote u H t r E = Ey x t j u r H = H x t k (, ) (, ) z Special case: plane waves satisfy the wave equation 2 2 ψ 1 ψ = 2 2 2 x c t With the known solution of the form ψ = Asin( ωt + φ)
Plane Electromagnetic Waves Electromagnetic field E y B z c x