30. 유도와유도용량 (Induction and Inductance) Faraday 법칙 dφ V 유도용량 (Inductance) NΦ i 유도기전력 (Induced emf) N dφ di : Self-Inductance
지난시간에 iot-savart aw Ampere s aw Force by currents Solenoid r r r μ0 ids d 3 4 π r r r ds μ0i enc F ba μ0 πd μ ni 0 d E i a i b 1 4π r r E da o dq r q encl o rˆ (Coulomb s aw) (Gauss aw)
30. 유도와유도용량 (Induction and Inductance) Faraday 의두가지실험 도선주위에서자석을움직여자기장을변화시킴 자석대신다른도선고리를두고, 전류를조절하여자기장을변화시킴 자기유도 (Magnetic induction) 유도전류 ( 유도기전력 ) 발생
유도효과 (induction effect) Inside the shaded region, there is a magnetic field into the board. If the loop is stationary, the orentz force predicts: r r r r F q( E+ v ) m (a) A Clockwise Current (b) A Counterclockwise Current (c) No Current When the loop is not moving, there is neither E nor v, so F m 0. no motion of charge and no current.
유도효과 (induction effect) Now the loop is pulled to the right at a velocity v. The orentz force will now give rise to: (a) A Clockwise Current (b) A Counterclockwise Current (c) No Current, due to Symmetry F m I v The charges in the wire all experience an upward force those in the end segment produce a clockwise current. v r r
유도효과 (induction effect) 1831: Michael Faraday did the previous experiment, and a few others: -v I Move the magnet, not the loop. Here there is no orentz force v r r but there was still an identical current!!! This coincidence bothered Einstein, and eventually led him to the Special Theory of Relativity (all that matters is relative motion). Decrease the strength of. Now nothing is moving, but M.F. still saw a current!! I d Decreasing I
30-3. Faraday 의유도법칙 도선고리를지나는자기력선의수가시간에따라변하면 고리에기전력 (: electromotive force) 이생긴다 Electric current is induced by the change of the Magnetic Flux Magnetic Flux Φ r da [Unit] weber (Wb), 1 Wb 1 T m dφ Faraday 법칙
30-3. Faraday 의유도법칙 고리를지나는자기력선의수를바꾸는방법 Φ r da 코일속자기장세기 를바꾼다. ( ) 자기장속에서의코일의넓이를바꾼다. ( A ) 자기장의방향에대한코일의방향을바꾼다. (θ). 위셋의복합. For N loops N dφ d ( Acosθ )
30-4. enz 법칙 The magnetic field due to the induced current has opposite sign to the change in the magnetic flux. 자기유도로생기는전류는자기다발의변화를방해하는방향으로흐른다. dφ
enz's aw and energy consideration enz's aw: The induced current will appear in such a direction that it opposes the change in flux that produced it. S N v N S S N N S v Conservation of energy considerations: Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated. Why??? If current reinforced the change, then the change would get bigger and that would in turn induce a larger current which would increase the change, etc..
전기기타 (Electric guitar) Toggle switch High frequency ow frequency 확인문제. 자기장이일정한비율로증가 / 감소할때, 고리에유도되는전류의크기가큰순서는? a b > c0
보기문제 30-3. t 0.10s 에서전류고리에생기는유도기전력의크기와방향은? 크기 : emf (t 0.10s) 14.4 V 방향 : 반시계방향
30-5. 유도와에너지전달 힘 F 가한일률 (power) Φ x d Φ dx v v i R R r r v F m i i R r r r r F F F F 1 3 F 일률 (Power): v R P Fv v R F (v 일정 ) 전류 i 에의한열에너지방출률 (power) P i R v R 전류고리를당길때힘 F 가한일은모두열에너지로변환된다.
MRI 주사중일어날수있는화상 ( 산소농도계 ) Magnetic resonance image(mri) 장치에서가해지는변화하는자기장 oop current 발생가능성있음!
소용돌이전류 (Eddy current, or Foucault current) http://www.n-ed.org/educationresources/highschool/electricity/eddycurrents.htm Roller coaster 의 braking Vending machines (detection of coins)
Applications of Magnetic Induction AC Generator Water turns wheel rotates magnet changes flux induces emf drives current Dynamic Microphones (E.g., some telephones) Sound oscillating pressure waves oscillating [diaphragm + coil] oscillating magnetic flux oscillating induced emf oscillating current in wire Question: Do dynamic microphones need a battery?
More Applications of Magnetic Induction Tape / Hard Drive / ZIP Readout Tiny coil responds to change in flux as the magnetic domains (encoding 0 s or 1 s) go by. Credit Card Reader Must swipe card generates changing flux Faster swipe bigger signal
More Applications of Magnetic Induction Magnetic evitation (Maglev) Trains Induced surface ( eddy ) currents produce field in opposite direction Repels magnet evitates train S N eddy current rails Maglev trains today can travel up to 310 mph Twice the speed of Amtrak s fastest conventional train! May eventually use superconducting loops to produce -field No power dissipation in resistance of wires!
30-6. 유도전기장 (Faraday 법칙 ) 변하는자기장은전류고리에유도전류를만든다 변하는자기장은전기장을만든다 dφ Assume the magnetic field changes in time 0 W W q 0 r r r r F ds q E ds 0 : 전하 q 0 가한바퀴돌때유도기전력이한일 : 전하 q 0 가한바퀴돌때유도전기장이한일 r E ds r r r E ds dφ Faraday s aw
Electro-Motive Force or emf time A magnetic field, increasing in time, passes through the blue loop An electric field is generated ringing the increasing magnetic field Circulating E-field will drive currents, just like a voltage difference oop integral of E-field is the emf r r E dl Note: The loop does not have to be a wire the emf exists even in vacuum! When we put a wire there, the electrons respond to the emf current. r r Note: In electrostatics E d l 0, so we could define a potential independent of path. This holds only for charges at rest (electrostatics). Forces from changing magnetic fields are nonconservative, and no potential can be defined!
정전기장 (static E) 과유도전기장 (Induced E) 이다른점 양 (+) 전하에서시작하여음 (-) 전하에서끝남. 경로에무관 : conservative force 닫힌고리에서는 V 0 임. 닫힌고리임에도불구하고변하는자가다발에서는 V0 이아님. 경로에의존 : nonconservative force 유도전기장에서는전기퍼텐셜을정의할수없다.
Escher depiction of nonconservative emf
확인질문 The magnetic field in a region of space of radius R is aligned with the z-direction and changes in time as shown in the plot. What is the direction of the induced electric field around a ring of radius R at time tt 1? (a) E is ccw (b) E 0 (c) E is cw x x x x x x x x x x x x x x x x x x R x x x x x x x x x x z y x There will be an induced emf at tt 1 because the magnetic field (and therefore the magnetic flux) is changing. t 1 t It makes NO DIFFERENCE that at tt 1 the magnetic field happens to be equal to ZERO! The magnetic field is increasing at tt 1 (actually at all times shown!) which induces an emf which opposes the corresponding change in flux. Electric field must be induced in a clockwise sense so that the current it would drive would create a magnetic field in the -z direction. Note: There does not need to be an actual wire the electric field exists anyway
확인질문 The magnetic field in a region of space of radius R is aligned with the z-direction and changes in time as shown in the plot. What is the relation between the magnitudes of the induced electric fields E R at radius R and E R at radius R? (a) E R E R (b) E R E R (c) E R 4E R x x x x x x x x x x x x x x x x x x R x x x x x x x x x x z y x t 1 t The rate of change of the flux is proportional to the area: E R dφ The path integral of the induced electric field is proportional to the radius: πr d r r E dl E( πr)
확인문제 4. r E ds r 경로 : (1), (), (3) 3, (4) 0 일때, 각구역에걸린자기장의방향은? (4) c 와 e 는반대방향, () d 와 e 는같은방향, (3) b 와 c 는 a 와같은방향 a : out b, c : out d, e : into
30-7. 유도기 (inductor) 와유도용량 (inductance) 유도기 (Inductor) An inductor is a two-terminal terminal device that consists of a coiled conducting wire wound around a core A current flowing through the device produces a magnetic flux φ forms closed loops threading its coils Total flux linked by N turns of coils, total flux is Φ Nφ For a linear inductor, Φ is proportional to i i Φ i Φ /i is the inductance + Unit: Henry (H) or (V s/a) v _ N Nφ
유도용량 (inductance) 인덕터의유도용량 (Inductance) NΦ i [1 Henry 1 H 1 T m /A] 솔레노이드의중심부근길이 l 인부분의자기다발 NΦ nl) A ( nl)( μ in) A NΦ μ 0 n la i μ 0 l ( 0 n A : 단위길이당유도용량
30-8. 자체유도 (Self-inductance) -field: μ 0 ni 자체유도기전력 ( ) : d( NΦ Nμ n 0 μ0n l ) di A A di l μ 0 n A dφ N di : Self-Inductance : Against the increase of the current
유도기의연결 Inductance in Parallel: i i 1 + i di1 1 di di i i 1 i 1 i di di 1 + di 1 1 1 1 + Inductance in Series: 1 1 + ( 저항연결인경우와같음 )
30-9. R 회로 i (a) 연결시 전류의증가 ir ir di di + Ri + 0 + 0 0 (b) 단절시 전류의감소 di + Ri 0 i (1 e R t / τ ), τ R i t / τ t / τ e i0e R R /R i 1 e 1 t / τ c e t / τ c t 1 t t
확인문제 6. (a) 스위치가닫힌직후전류가큰순서는? (b) 오랜시간후전류가큰순서는? (a) > 3 > 1 (0) (b) > 3 > 1
30-10. 자기장에저장된에너지 In R Circuit ir + Power: di i i R + i di + R Inductor 의일률 du i di U 1 i Cf ) Energy stored in a capacitor UC 1 CV
30-11. 자기장의에너지밀도 In a solenoid (Magnetic Energy) μ 0 n l A l A U 1 μ0 ( n l A) i μ ni i 0 μ n 0 U l A V0 l μ0 μ0 u μ 0 (Magnetic energy density) Cf) u E 1 0E (Electric energy density)
How we can store the energy? Energy stored in a capacitor... E E ( t) 1 Cv ( t) energy density u + + + + + + + + dielectric - - - - - - - - 1 E electric 0 E Energy stored in an inductor. E M ( t) 1 i ( t) energy density... u magnetic 1 μ 0
30-1. 상호유도 (Mutual induction) 상호유도용량 (mutual inductance) : M 전류 i 1 이흐르는코일 -1 에의한코일 - 의상호유도용량 M M 1 NΦ i 1 1 di dφ 1 1 1 i1 NΦ 1 M 1 N 전류 i 이흐르는코일 - 에의한코일 -1 의유도용량 1 di 1 M 1 di M1 M 1 M1 M 1 di di M M 1
30. Summary r E r ds dφ Faraday s aw (enz s aw) Eddy current μ 0 l n A : 단위길이당유도용량 dφ N di : Self-Inductance : Against the increase of the current Magnetic energy 1 U i u μ 0