Overview of Electromagnetic Fields 2

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1 DR. GYURCSEK ISTVÁN Overview of Eectromagnetic Fieds 2 Magnetic Fied, Couped Fieds Sources and additiona materias (recommended) Dr. Gyurcsek Dr. Emer: Theories in Eectric Circuits, GobeEdit, 2016, ISBN: Ch. Aexander, M. Sadiku: Fundamentas of Eectric Circuits, 6th Ed., McGraw Hi NY 2016, ISBN: Simonyi K.: Viamosságtan. AK Budapest 1983, ISBN: Dr. Semeczi K. Schnöer A.: Viamosságtan 1. MK Budapest 2002, TK szám: 49203/I Dr. Semeczi K. Schnöer A.: Viamosságtan 2. TK Budapest 2002, ISBN: Zombory L.: Eektromágneses terek. MK Budapest 2006, ( 1 gyurcsek.istvan@mik.pte.hu

2 Magnetic Fied Sources 2 gyurcsek.istvan@mik.pte.hu

3 Stationary Magnetic Fied Couped (Time Varying) Fieds 3 gyurcsek.istvan@mik.pte.hu

Static and Stationary MF Static MF MF of permanent magnet (ferromagnetic body) Stationary magnetic fied MF of DC Magnetic fied intensity (H) (Ampere s exc.

4 Static and Stationary MF Static MF MF of permanent magnet (ferromagnetic body) Stationary magnetic fied MF of DC Magnetic fied intensity (H) (Ampere s exc. aw, see ater) H r = I 2π r H(r) = I 0 r 0 2π r 0 unit vector with the direction of current r 0 unit vector with the direction of radius Magnetic permeabiity interaction bw. materia (or vacum) and MF μ Vs, μ = μ Am 0 μ r, μ 0 = 4π 10 7 Vs Am Magnetic induction (magn. fux density, B) B = μ H Magnetic induction fux (magn. fux, φ) Φ = න B da A 4 gyurcsek.istvan@mik.pte.hu

5 Gauss Law of Magnetism (Maxwe s 3rd eq.) Integra form Visuaization of magnetic fied ර B da = 0 Differentia form div B = 0 B cury vector fied. 5 gyurcsek.istvan@mik.pte.hu

Ampére s Force Law (Q, v) Lorentz s force aw F = Q v

eectric current (A) The ampere is that constant

conductors of infinite ength, of negigibe circuar

produce between these conductors a force equa to 2 x

6 Ampére s Force Law (Q, v) Lorentz s force aw F = Q v B Ampére s force aw F = μ I 1 I 2 2π r SI unit of eectric current (A) The ampere is that constant current which, if maintained in two straight parae conductors of infinite ength, of negigibe circuar crosssection, and paced 1 meter apart in vacuum, woud produce between these conductors a force equa to 2 x 10 7 N per meter of ength. 6 gyurcsek.istvan@mik.pte.hu

Ampere s Excitations Law André-Marie Ampère (1775

7 Ampere s Excitations Law André-Marie Ampère ( ) [physics], [chemistry], [maths] (One of the 72 names inscribed on the Eiffe Tower.) N ර H d = න J da ර H d = i i A i=1 7 gyurcsek.istvan@mik.pte.hu

8 Ampere s Excitations Law Exampe 1 Magnetic fied in soenoids (ong cois) Exampe 2 Magnetic fied in toroids ර H d + ර 0 d = N I in out Int. homogeneous fied H = N I H = N I B = μ N I Same Way B = μ N I 2πR 8 gyurcsek.istvan@mik.pte.hu

Energy stored in an inductor p t = u t i t = L t w(t) = න L di t dt di t dt i t dt = L di t u

9 Energy of Magnetic Fied (Maxwe s 6th Eq.) w M = 1 2 H B = 1 2 μ H2 = 1 2 B2 μ Energy density W = 1 2 න V H B dv W w M = im V 0 V Energy stored in an inductor p t = u t i t = L t w(t) = න L di t dt di t dt i t dt = L di t u t = L dt i t w(t) = න p t dt i(t) න i( ) (see Faraday) t i(t) i t di = i2 2 i( ) i(t) ቚ = 0 W = 1 t= 2 L i2 (t) 9 gyurcsek.istvan@mik.pte.hu

10 Interaction of Matter Magnetic dipoes w/o. and w. magnetic fied Diamagnetic materias μ r < 1, however μ r B (magnetic fux density) of an externa H (magnetic fied intensity) ower than it was in vacuum Paramagnetic materias 1 < μ r < 10 2 B (magnetic fux density) of an externa H (magnetic fied intensity) higher than it was in vacuum Ferromagnetic materias μ r > 10 2 B (magnetic fux density) of an externa H (magnetic fied intensity) much higher than it was in vacuum 10 gyurcsek.istvan@mik.pte.hu

11 The Hysteresis Loop (Ferromagnetic Materias)

12 Sef and Mutua Inductance Coi Fux H = N I Φ = μ N I Ψ = N Φ L Ψ I = μ N2 A, B = μ H, Φ = B A A Ψ = μ N2 A I L 21 = Ψ 21 I 1, L 21 = L 12 = M Couping factor (0 1) Idea (tight) couping k = M L 1 L 2 M = L 1 L 2 12 gyurcsek.istvan@mik.pte.hu

13 Stationary Magnetic Fied Couped (Time Varying) Fieds

14 Genera Form of Ampere s Law Ampere s excitation aw (precondition I = constant!) ර H = න J S = I modified by Maxwe (I time varying) ර H = I ර S 1 S 2 H = I d ර H = න J + J d S = I + I d I conducting current (carried by the wire) I d dispacement current 15 gyurcsek.istvan@mik.pte.hu

15 Ampere s Law with Maxwe s Ext. (Maxwe s 1st Eq.) Integra form ර H d = න A J + D t da Differentia form rot H = J + D t 16 gyurcsek.istvan@mik.pte.hu

16 Maxwe-Faraday Law of Induction (Maxwe s 2nd Eq.) U = N dφ dt Integra form ර E d = t න A B da Differentia form rot E = B t 17 gyurcsek.istvan@mik.pte.hu

17 Exampes for Maxwe s 2nd Eq. Exampe 1 Eddy current ර E d = t න A B da Exampe 2 Inductor Characteriscics This (cury!) EF acceerates charges over a circe Eddy current in cosed wire (conducting materia) L Ψ t i t i(t) = μ N2 A u t = dψ(t) dt = u t = L di(t) dt (see sef inductance) d L i(t) dt 18 gyurcsek.istvan@mik.pte.hu

18 Maxwe s Equations (Integra Forms Differentia Forms) (1) Ampere excit. ර H d = න A J + D t da rot H = J + D t (5) materias D = ε E, B = μ H, J = σ E (2) Faraday induct. ර E d = t න A B da rot E = B t (6) energy w EM = 1 2 E D + B H (3) Gauss magnet. (4) Gauss e. stat. ර A ර A B da = 0 div B = 0 D da = න V ρ dv div D = ρ H = J + D t, B = 0 E = B t, James Cerk Maxwe ( ) D = ρ 19 gyurcsek.istvan@mik.pte.hu

19 Questions

Circuit Theorems DR. GYURCSEK ISTVÁN

Circuit Theorems DR. GYURCSEK ISTVÁN DR. GYURCSEK ISTVÁN Circuit Theorems Sources and additional materials (recommended) q Dr. Gyurcsek Dr. Elmer: Theories in Electric Circuits, GlobeEdit, 2016, ISBN:978-3-330-71341-3 q Ch. Alexander, M.