Chapter 32 Inductance

Transcription

1 Chapter 3 nductance 3. Sef-nduction and nductance Sef-nductance Φ BA na --> Φ The unit of the inductance is henry (H). Wb T H A A When the current in the circuit is changing, the agnetic fux is aso changing. ( ) dφ d --> The induced ef shoud be ε dφ The sef inductance of a infinite ong soenoid: N N Φ NAB NA A --> ε Φ N A Considering the inductor having an interna resistance r, the potentia difference is: V ε r r Exape: Mode a ong coaxia cabe as two thin, concentric, cyindrica conducting shes of radii a and b and ength. The conducting shes carry the sae current in opposite directions. Cacuate the inductance of this cabe. Cacuate agnetic fux: b b B Φ πr drdz n πr π a a b Φ n π a

2 3. Circuits Use Kirchhoff s rue: E r d r ε Differentia Eq: ε --> ε --> ( ) ε --> ( ε ) ( ε ) d ε --> n ε --> ε ε e --> ε e Tie constant: τ Exape: Find the tota energy dissipated in the resistor, when the current in the inductor decreases fro its initia vaue of to? ε e, P --> U t e 3.3 Energy in a Magnetic Fied Obtain the agnetic energy fro the ef induced by sef inductance. Φ --> The induced ef is dφ ε The energy dissipated or the power is P V ε The tota energy when the current has reached its fina vaue f is: U t tf f t Cacuate the agnetic energy by obtaining the energy stored in the sef inductor of an infinite soenoid.

3 B n, na ( n ) U Φ n u n ( A) u V B <----> ε u e E (Do you reeber how to get this?) Exape: A certain region of space contains a unifor agnetic fied of. T and a unifor eectric fied of.5 X 6 N/C. Find (a) the tota eectroagnetic density. u e ε 6 3 ( 8.85 )(.5 ) 7.7 E J/ u (.) B 7 ( 4π ) 59J/ Mutua nductance Mutua nductance The agnetic fied of oop is: B ~ The fux at is ~ π Φ π M π The fux at is Φ ~ M oop oop The concept of inductance: Φ M M M --> The utua inductance is deterined when the geoetrica configuration between the two oops is given. N B in due to : B N Fux in : Φ N ( π r ) M N B in due to : B N Fux in : Φ N ( π r ) M 3

4 ( r ) NN M M π 3.5 Osciations in an C Circuit Kirchhoff s ue: d + C C Copare with: d x + kx F a kx soution: x( t) Acos ( ω t +φ) d + C ω sin ωt + φ soution: ( t) cos( ω t +φ) ( ) ax ax, ω U UC + U + ax cos ax φ C C ax C C ( ωt + φ ) + ω sin ( ωt + ) 4

5 Appy the Kirchhoff s oop rue: C d oop direction --> + ( nd Differentia Equation, copare with C haronic osciation: where k ω ) F d x a kx with the answer of A ( ω t + δ ) x cos, eeber the pattern of this differentia equation: d + --> the soutions are periodica functions and the useabe functions C x x ix exp x ), are sin ( ) ( cos ( )), exp ( ) ( ( ) Guess that the answer is A ( Bt + C) initia conditions.) --> cos( ) cos( ) cos. (Here A and C can be deterined by B A Bt + C + A Bt + C --> B ω C C d --> Acos ( ωt δ ) & ωasin( ωt δ ) sin( ωt) Capacitor --> Eectric Fied --> Potentia Energy nductor --> Moving of Charges --> Kinetic Energy 5

6 The average energy stored in the capacitor (inductor) is ( ). C The instantaneous energy transferring in the circuit is: C A cos C ( ωt δ ) + + ω A sin δ A C ( ωt ) C What are physica pictures of and? Exape: A -F capacitor is charged to V and the capacitor is then connected across a 6-H inductor. (a) What is the frequency of osciation? (b) What is the vaue of the current? (a) ω, (b) C CV C Sipe AM adio receiver: 3.6 The C Circuit Kirchhoff s ue: C Power Consideration: P + C d d + + C Copare with daped osciation: d x dx + b + kx nt ( t) Ae x, n + bn + k n b ± b 4k b 4k < x( t) Ae b cos 4k b t ( ) t ax e cos 4 / C t 6

7 C Circuit (Daped Osciation) Diff Eq: C d d --> + + (Daped Osciations: F a kx bv ) C The natura frequency (no resistaor) is: We guess a soution of d d + +. C oop direction ω C ( d + ). C Bt Ae for soving the differentia equation Bt B + B + Ae --> C ± B C 4 ± C over-daped: > C 7

8 under-daped: < C under-daped soution: Ae t ± i e ω t The energy distributed in the circuit eeents is: d d d > + + C C d --> + + C 8