Page 1 Magnetic Materials • The inductor F B = LI (Q = CV) 1 ?B ?F B = L ?I ?×E = - (CGS) ?t ?t ?×EdS c = ?t - 1 ? ( BdS )= - 1 ?F V EMF = - ?N ? F t B = -L ? ? I t ?? c ?t ?? c ?t B ?I ?V F B = magnetic flux density V = L (recall I =C for the capacitor) ?t ?t ?? ?×EdS = ? E ·d l (Green's Theorem) ?I Power =VI = LI V = ? E ·d l = - 1 ?F B (explicit Faraday's Law) ?t c ?t Energy = ? Power ·dt = ? LIdI = 1 LI 2 = 1 NF B I 2 2 ? 1 2 ? ? capacitor CV ? ? 2 ? ©1999 E.A. Fitzgerald 1 Page 2 Magnetic Materials • The inductor F B = LI (Q = CV) 1 ?B ?F B = L ?I ?×E = - (CGS) ?t ?t ?×EdS c = ?t - 1 ? ( BdS )= - 1 ?F V EMF = - ?N ? F t B = -L ? ? I t ?? c ?t ?? c ?t B ?I ?V F B = magnetic flux density V = L (recall I =C for the capacitor) ?t ?t ?? ?×EdS = ? E ·d l (Green's Theorem) ?I Power =VI = LI V = ? E ·d l = - 1 ?F B (explicit Faraday's Law) ?t c ?t Energy = ? Power ·dt = ? LIdI = 1 LI 2 = 1 NF B I 2 2 ? 1 2 ? ? capacitor CV ? ? 2 ? ©1999 E.A. Fitzgerald 1 The Inductor 4p 1 ? E ?× B = J + c c ? t 4p 4p ?? ?× BdS = ? B · d l = ?? J · dS = I c c 4p B = In c N = n · length = nl Nf B N (BA) 4p 2 L = = = n lA I I c Insert magnetic material Magnetic dipoles in material can line-up in magnetic field B = H + 4p? H = H + 4p M B magnetic induction ? M ? magnetic susceptibility M = ?H = ? µ = 1+ 4p? H magnetic field strength (applied field) ? H M magnetization B = 4pM + 1 B = µH 2 ©1999 E.A. Fitzgerald Page 3 Magnetic Materials • The inductor F B = LI (Q = CV) 1 ?B ?F B = L ?I ?×E = - (CGS) ?t ?t ?×EdS c = ?t - 1 ? ( BdS )= - 1 ?F V EMF = - ?N ? F t B = -L ? ? I t ?? c ?t ?? c ?t B ?I ?V F B = magnetic flux density V = L (recall I =C for the capacitor) ?t ?t ?? ?×EdS = ? E ·d l (Green's Theorem) ?I Power =VI = LI V = ? E ·d l = - 1 ?F B (explicit Faraday's Law) ?t c ?t Energy = ? Power ·dt = ? LIdI = 1 LI 2 = 1 NF B I 2 2 ? 1 2 ? ? capacitor CV ? ? 2 ? ©1999 E.A. Fitzgerald 1 The Inductor 4p 1 ? E ?× B = J + c c ? t 4p 4p ?? ?× BdS = ? B · d l = ?? J · dS = I c c 4p B = In c N = n · length = nl Nf B N (BA) 4p 2 L = = = n lA I I c Insert magnetic material Magnetic dipoles in material can line-up in magnetic field B = H + 4p? H = H + 4p M B magnetic induction ? M ? magnetic susceptibility M = ?H = ? µ = 1+ 4p? H magnetic field strength (applied field) ? H M magnetization B = 4pM + 1 B = µH 2 ©1999 E.A. Fitzgerald H and B • H has the possibility of switching directions when leaving the material; B is always continuous 3 ©1999 E.A. Fitzgerald p q H B At p: At q: B H M=0 M M M H B Page 4 Magnetic Materials • The inductor F B = LI (Q = CV) 1 ?B ?F B = L ?I ?×E = - (CGS) ?t ?t ?×EdS c = ?t - 1 ? ( BdS )= - 1 ?F V EMF = - ?N ? F t B = -L ? ? I t ?? c ?t ?? c ?t B ?I ?V F B = magnetic flux density V = L (recall I =C for the capacitor) ?t ?t ?? ?×EdS = ? E ·d l (Green's Theorem) ?I Power =VI = LI V = ? E ·d l = - 1 ?F B (explicit Faraday's Law) ?t c ?t Energy = ? Power ·dt = ? LIdI = 1 LI 2 = 1 NF B I 2 2 ? 1 2 ? ? capacitor CV ? ? 2 ? ©1999 E.A. Fitzgerald 1 The Inductor 4p 1 ? E ?× B = J + c c ? t 4p 4p ?? ?× BdS = ? B · d l = ?? J · dS = I c c 4p B = In c N = n · length = nl Nf B N (BA) 4p 2 L = = = n lA I I c Insert magnetic material Magnetic dipoles in material can line-up in magnetic field B = H + 4p? H = H + 4p M B magnetic induction ? M ? magnetic susceptibility M = ?H = ? µ = 1+ 4p? H magnetic field strength (applied field) ? H M magnetization B = 4pM + 1 B = µH 2 ©1999 E.A. Fitzgerald H and B • H has the possibility of switching directions when leaving the material; B is always continuous 3 ©1999 E.A. Fitzgerald p q H B At p: At q: B H M=0 M M M H B Maxwell and Magnetic Materials • Ampere’s law ? H · d l = I = 0 • For a permanent magnet, there is no real current flow; if we use B, there is a need for a fictitious current (magnetization current) • Magnetic material inserted inside inductor increases inductance F B = BA ~ 4pMA = 4p?HA = 4p? ? ? 4p In ? ? A ? c ? NF B () 2 Material Type ? 4p 2 L = = n lA? I c Paramagnetic +10 -5 -10 -4 L increased by ~? due to magnetic material Diamagnetic -10 -8 -10 -5 Ferromagnetic +10 5 ©1999 E.A. Fitzgerald 4 Page 5 Magnetic Materials • The inductor F B = LI (Q = CV) 1 ?B ?F B = L ?I ?×E = - (CGS) ?t ?t ?×EdS c = ?t - 1 ? ( BdS )= - 1 ?F V EMF = - ?N ? F t B = -L ? ? I t ?? c ?t ?? c ?t B ?I ?V F B = magnetic flux density V = L (recall I =C for the capacitor) ?t ?t ?? ?×EdS = ? E ·d l (Green's Theorem) ?I Power =VI = LI V = ? E ·d l = - 1 ?F B (explicit Faraday's Law) ?t c ?t Energy = ? Power ·dt = ? LIdI = 1 LI 2 = 1 NF B I 2 2 ? 1 2 ? ? capacitor CV ? ? 2 ? ©1999 E.A. Fitzgerald 1 The Inductor 4p 1 ? E ?× B = J + c c ? t 4p 4p ?? ?× BdS = ? B · d l = ?? J · dS = I c c 4p B = In c N = n · length = nl Nf B N (BA) 4p 2 L = = = n lA I I c Insert magnetic material Magnetic dipoles in material can line-up in magnetic field B = H + 4p? H = H + 4p M B magnetic induction ? M ? magnetic susceptibility M = ?H = ? µ = 1+ 4p? H magnetic field strength (applied field) ? H M magnetization B = 4pM + 1 B = µH 2 ©1999 E.A. Fitzgerald H and B • H has the possibility of switching directions when leaving the material; B is always continuous 3 ©1999 E.A. Fitzgerald p q H B At p: At q: B H M=0 M M M H B Maxwell and Magnetic Materials • Ampere’s law ? H · d l = I = 0 • For a permanent magnet, there is no real current flow; if we use B, there is a need for a fictitious current (magnetization current) • Magnetic material inserted inside inductor increases inductance F B = BA ~ 4pMA = 4p?HA = 4p? ? ? 4p In ? ? A ? c ? NF B () 2 Material Type ? 4p 2 L = = n lA? I c Paramagnetic +10 -5 -10 -4 L increased by ~? due to magnetic material Diamagnetic -10 -8 -10 -5 Ferromagnetic +10 5 ©1999 E.A. Fitzgerald 4 Microscopic Source of Magnetization • No monopoles • magnetic dipole comes from moving or spinning electrons Orbital Angular Momentum A µ µ is the magnetic dipole moment r r Energy = E = -µ ·H = - µ H cos? I e- L What is µ? For ?=0, E = -µH ˜ -F B I since energy ~ LI 2 and for 1loop L = F B I F B = ?? H ·dS ~ HA ?µH = F B I = HAI and ?µ = IA I = - e ? A = pr 2 c 2p µ = - e ?r 2 2c 5 ©1999 E.A. FitzgeraldRead More

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