# Lesson 13 Inductance, Magnetic energy /force /torque

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1 Lesson 3 nductnce, Mgnetic energy /force /torque 楊 尚 達 Shng-D Yng nstitute of Photonics Technologies Deprtment of Electricl Engineering Ntionl Tsing Hu Uniersity, Tiwn

2 Outline nductnce Mgnetic energy Mgnetic force Mgnetic torque

3 Sec. 3- nductnce. Self & mutul inductnces. Elution procedures

4 Definition- Closed loop C crrying current will crete Φ B ds, flux: flux linge: B S f, by r B Λ μ dl 4π ' C B NΦ B rb, Φ B ds rφ, Λ rλ Self-inductnce of the loop C : S L Only depend on geometry Λ

5 Definition- n the presence of nother loop C, will pss through C, mutul flux linge: where Φ B ds S B B L B Λ N Φ Mutul-inductnce between the loops: Λ Depend on geometry & mteril.

6 Comment μ dl A( r), ' 4π C ( r, r ) L NΦ N S B A μ N ( ) A ds A L dl 4π C N C A dl μnn dl dl 4π C C μnn dl dl 4π C C L L

7 Elution of inductnce (Method ). Assume current flowing on the loop.. Find B by Ampere s lw or Biot-Srt lw: μ dl H dl, B ' C C 4π 3. Find Λ( ) by Λ N B ds 4. Find L by L Λ, independent of S

8 Elution of inductnce-reference figure B C μ 4π dl dl

9 Elution of inductnce (Method ). Assume current flowing on the loop.. Find H nd B by Method 3. Find the stored energy W m V ( ) H B d 4. Find L by Wm L

10 Exmple 3-: Solenoid inductor () Consider hollow solenoid with cross-sectionl re S, n turns per unit length. Find the inductnce per unit length L.. Assume current flowing on the loop.. By Ampere s lw: μ n B

11 Exmple 3-: Solenoid inductor () 3. For unit length (l), 4. By definition: Λ Λ n Φ n n ( n ) ( μ n ) S μ S n μ S L

12 Exmple 3-: Two concentric coils () Consider two coils C, C with N, N turns nd lengths l, l. They re wound concentriclly on thin cylindricl core of rdius with permebility μ. Find the mutul inductnce L.. Assume C, C he currents,

13 Exmple 3-: Two concentric coils () N. By Ampere s lw, uniform field B μ l 3. Flux linge of C due to C : Λ N Φ N B S 4. By definition: L N N l μ π

14 Sec. 3- Mgnetic Energy. Energy of ssembling current loops. Energy of mgnetic fields

15 Energy of ssembling current loops-one loop () Closed loop C with self-inductnce L. f the loop current i increses from to slowly (qusi-sttic), n emf of: dφ dt L di dt will be induced on C to oppose the chnge of i (Frdy s lw, Lenz s lw).

16 Energy of ssembling current loops-one loop () The wor done to oercome the induced nd enforce the chnge of i is: W di dt ( t) i ( t) dt L idt L i di L which is stored s mgnetic energy: W one loop L

17 Energy of ssembling current loops-two loops () nsert loop C with self-inductnce L, mutul inductnce L. f we mintin i, while i increses from to slowly, n emf of: dφ di L dt dt will be induced on C in n ttempt to chnge i wy from

18 Energy of ssembling current loops-two loops () The wor done to mintin i is: W di dt t) dt L dt L ( di L

19 Energy of ssembling current loops-two loops (3) Menwhile, n emf of: dφ di L dt dt will be induced on C to oppose the chnge of i (from to ). The wor done to oercome nd enforce the chnge of i is: W L

20 Energy of ssembling current loops-two loops (4) The totl mgnetic energy stored in the system of two current loops is: W L + L + two loops L

21 Energy of ssembling current loops-n loops The totl mgnetic energy stored in the system of N current loops crrying currents,,.., N, is: W m N N j By L Λ, the flux (linge) of loop C due to ll the N current loops: Φ N j L j j, L j j N W m Φ

22 Energy of continuous current distributions- Decompose system of continuous current distribution J (r ) in olume V' into N elementry current loops C, ech hs current Δ nd filmentry cross-sectionl re Δ B, A Φ C B ds S A dl

23 Energy of continuous current distributions- Δ Φ N C N m dl A W ( ) ( ) J dl J dl J dl Δ Δ Δ Δ, Δ N C m J A W ( ) V m d J A W

24 Comments W e N Q V N W m Φ W e V ( ρv ) d W m V ( A J ) d Electrosttics Mgnetosttics

25 Energy of mgnetic fields- n rel pplictions (especilly electromgnetic wes), sources re usully fr wy from the region of interest, only the fields re gien source O J (r ) H, B

26 Energy of mgnetic fields- () W m J (r ) contin ll the source currents V V V J H ( A J ) d A ( H ) () W By ector identity: A H H H B A H m d ( ) ( A) A ( H ) ( ) V V ( H B) d ( A H ) d

27 Energy of mgnetic fields-3 Q A ds A d, S V ( ) ( A H ) d ( A H ) V S ds J (r ) V S (3) W m ( ) ( ) H B d A H ds V S

28 Energy of mgnetic fields-4 S S Obs. pt. H A

29 Energy of mgnetic fields-5 ) m (J 3 B H energy density ( ) 4 ) ( ) ( H A ds H A S π V m m d r w W ) (

30 Energy of mgnetic fields-6 W m V H B d V dw m H B d

31 Exmple 3-3: Coxil cble inductor () Cylindricl symmetry, Ampere s lw, Find the stored mgnetosttic energy nd inductnce per unit length of: μ B H < < < b r r r r B if, if, π μ π μ φ φ

32 Exmple 3-3: Coxil cble inductor () Energy density: < < < b r r r r B H w m, 8, 8 4 π μ π μ dr r d π Differentil olume (L):

33 Exmple 3-3: Coxil cble inductor (3) Totl stored energy: r dr r W m <, π μ π μ b r b dr r W b m < <, ln 4 4 π μ π μ, L W m + + b W W L m m ln 8 ) ( π μ π μ internl externl

34 Sec. 3-3 Mgnetic Force. Force on current loops. Exmple: force between prllel wires

35 Force on current-crrying loops- Consider n elementl current-crrying wire of cross-sectionl re S, represented by differentil displcement ector dl Free chrges within the wire of chrge density ρ moe with elocity u( // dl ), experiencing force of: F ρs dl u B d m ( )

36 Force on current-crrying loops- dl u u dl J ρu df m ρs dl ρs u dl B df m ( u B) JS ( dl B) ( dl B) For current loop C : F m dl B C

37 Force on current-crrying loops-3 f B is creted by nother closed loop C crrying current, the force exerted on the loop C crrying current is: F B B dl B C μ μ dl 4π ' C 4π C dl

38 Force on current-crrying loops-4 F μ C C 4π dl ( dl ) F Counterprt in electrosttics: Coulomb s force between two chrges F q q 4πε

39 Exmple 3-4: Force between two long wires Find the force per unit length between two infinitely long, prllel wires seprted by d, crrying currents, in the sme direction. B μ x, F dl πd B y ( dz) z μ πd x μ πd ttrction force

40 Sec. 3-4 Mgnetic Torque. Exmple: mgnetic force & torque exerted on current loop

41 Exmple 3-5: Force & torque on current-crrying loops () Consider circulr loop on the xy-plne with rdius b, current in clocwise sense, nd is plced uniform mgnetic filed: B B + z B B y B

42 Exmple 3-5: Force & torque on current-crrying loops () The force exerted on differentil current element on the loop due to : dl φ bdφ B b φ df m df r bb ( dl B ), ( bdφ ) ( B ) φ dφ z

43 Exmple 3-5: Force & torque on current-crrying loops (3) df The force exerted on differentil current dl φ bdφ B : element on the loop due to df m bb d bb z ( dl B ), ( ) ( ) φbdφ yb φ( sinφ cosφ) ( ) x sinφdφ y y

44 Exmple 3-5: Force & torque on current-crrying loops (4) The totl force exerted on the loop due to B : df rbb dφ, φ F bb π df π r ( φ) dφ

45 Exmple 3-5: Force & torque on current-crrying loops (5) df The totl force exerted on the loop due to B : bb sinφd, z φ F z π bb df π sinφ dφ

46 Exmple 3-5: Force & torque on current-crrying loops (6) The totl torque exerted on the loop due to : B dφ, bb F d r φ ( ) π π r r r B b b df T

47 Exmple 3-5: Force & torque on current-crrying loops (7) The totl torque exerted on the loop due to : df bb sinφd, z φ B π T df rb b b B B φ π π [ ( )] z ( ) φ x sinφ ( ) r sinφdφ y cosφ sinφdφ

48 Exmple 3-5: Force & torque on current-crrying loops (7) T π π b B x sin φdφ y sinφ cos x ( πb ) B mb T T + T m ) B n generl, x cosφ T x ( z y sin φ φ dφ T m B

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