Electricity and Magnetism Energy of the Magnetic Field Mutual Inductance

Transcription

1 Electricity and Magnetism Energy of the Magnetic Field Mutual Inductance Lana Sheridan De Anza College Mar 14, 2018

2 Last time inductors resistor-inductor circuits

3 Overview wrap up resistor-inductor circuits energy stored in an inductor coaxial inductor mutual inductance

4 Reminder: Inductors A capacitor is a device that stores an electric field as a component of a circuit. inductor a device that stores a magnetic field in a circuit It is typically a coil of wire.

5 Reminder: Inductance inductance the constant of proportionality relating the magnetic flux linkage (NΦ B ) to the current: NΦ B = L I ; L = NΦ B I Φ B is the magnetic flux through the coil, and I is the current in the coil.

6 fter RLswitch After Circuits: Sswitch 1 is thrown S 1 current is thrown closed closed rising t t 0, at t the current 0, the current increases increases oward toward its maximum its maximum value value /R. Current e/r. in loop: The time The rate time of change rate of of change of current is current a maximum is a maximum at t 0, at t which is the which instant is the at instant which at which Derivative switch S 1 of switch is current: thrown S 1 is closed. thrown closed. e R e e R e.632 R0.632 R i i t L R t L R e L di dt e L di dt t t t t ure 32.3 Plot of the current Figure 32.3 Plot of the current versus i(t) time = for E (1 the RL e Rt/L) circuit R sus time for the RL circuit own in Figure The time nstant shown t is the in Figure time interval The time uired constant for i to t reach is the 63.2% time interval of its ximum required value. for i to reach 63.2% of its maximum value. t t Figure 32.4 di Plot of di/dt versus time for Figure the RL circuit 32.4 shown Plot of in di/dt Fig- versu dt = E L e Rt/L ure time The for rate the decreases RL circuit exponentially with ure time as i The increases rate decreases toward exp shown in its maximum tially value. with time as i increases tow its maximum value.

7 or RL based Circuits: on Eq. current 30-35, and The resistor's potential rising tion V ir. llows from difference Eq turns If on. we The inductor's potential Potential difference drop across turns resistor: off.. V R (V) V L (V) (30-43) rent in the circuit to reach the potential difference V R a graph of the increasing Fig a nough for the equilibrium t (ms) the effect will be (a) to remove must actually Vbe R = made ir an h that does this is called a e current through the resisy to zero but must decay to V R t (ms) emf across inductor: (a) V L (V) t (ms) (b) E L (t) = L di Fig The variation dt with time o (a) V R,the potential difference across th resistor in the circuit of Fig , and ( V L,the potential difference across the in

8 RL Circuits: Time Constant τ L = L/R τ L is called the time constant of the circuit. (Notice that it is defined differently for RL circuits as opposed to RC circuits.) This gives the time for the current to reach (1 e 1 ) = 63.2% of its final value. Alternatively, it is the time for the potential drop across the inductor to fall to 1/e of its initial value. It is useful for comparing the relaxation time of different RL-circuits.

9 ccurs if we introduce ining RLa resistor Circuits: R and current falling for example,the curpresent, the current ctor,however,a selff opposes the rise of in polarity. Thus, the o emfs, a constant induction. As long as + a b S R L s becomes less rapid Fig An RL circuit. When switch Battery switched out - switch to b. roportional to di/dt, S is closed on a,the current rises and approaches a limiting value /R. s /R asymptotically. Loop rule: E L ir = 0 Solution: L di dt ir = 0 i = E R e t/τ L ; τ L = L R

10 At t 0, the switch is thrown to RL Circuits: current position falling b and the current has (32.10) the instant mmediately present, it e exponenws that the e R its maximum value e/r. i t S 2 at posicross which resistor ng time, Figure 32.5 Current versus time for the right-hand loop of i = E the circuit R e t/τ L ; τ shown in Figure L = L R32.2. For t, 0, switch S 2 is at position a.

11 Energy Stored in a Magnetic Field in an Inductor Power delivered to inductor: P = I E L Energy stored in inductor carrying current i: U B = P dt = i 0 ( ) i L di dt dt U B = 1 2 Li 2 Compare with U E = Q2 2C (or U E = 1 2 CV 2 ) for the energy stored in a capacitor.

12 Energy Density of a Magnetic Field The energy density of a magnetic field is the energy stored per unit volume. u B = U B Al where A is the cross-sectional area of a solenoid and l is the length. u B = L i 2 Al 2 Remember, L = µ 0 n 2 Al and B = µ 0 in.

13 Energy Density of a Magnetic Field u B = B2 2µ 0 Compare with u E = 1 2 ɛ 0E 2 for electric fields.

14 Energy Density of a Magnetic Field Question Which of the following adjustments increases the energy density in a solenoid? (A) increasing the number of turns per unit length on the solenoid (B) increasing the cross-sectional area of the solenoid (C) increasing the current in the solenoid (D) two of the above

15 Energy Density of a Magnetic Field Question Which of the following adjustments increases the energy density in a solenoid? (A) increasing the number of turns per unit length on the solenoid (B) increasing the cross-sectional area of the solenoid (C) increasing the current in the solenoid (D) two of the above

16 Coaxial Cable Inductor (Ex 32.4) Model a long coaxial cable as a thin, cylindrical conducting shell of radius b concentric with a solid cylinder of radius a. The conductors carry the same current i in opposite directions. such as your video odel a long coaxial entric with a solid e same current I in of this cable. ve a visible coil in le such as the light rs are connected at presents one large field between the e current changes, original change in i dr Calculate the inductance Figure 32.7 L of(example a length32.4) l, ofsec- tion of a long coaxial cable. The this cable. inner a S B i r b

17 Coaxial Cable Inductor (Ex 32.4) Need to relate the flux linkage, NΦ B, to the current. (N = 1.) such as your video odel a long coaxial entric with a solid e same current I in of this cable. dr a i b ve a visible coil in le such as the light rs are connected at presents one large field between the e current changes, original change in must return to the i S B Φ Figure 32.7 (Example 32.4) Sec- B = B da, and area of rectangle is A: tion of a long coaxial cable. The inner and outer conductors carry equal currents in opposite da directions. = l dr r

18 Coaxial Cable Inductor (Ex 32.4) Need to relate the flux linkage, NΦ B, to the current. (N = 1.) Φ B = B da = b a = µ 0il 2π µ 0 i 2πr l dr b a = µ 0il 2π ln 1 r dr ( ) b a

19 Coaxial Cable Inductor (Ex 32.4) Need to relate the flux linkage, NΦ B, to the current. (N = 1.) Φ B = B da = b a = µ 0il 2π µ 0 i 2πr l dr b a = µ 0il 2π ln 1 r dr ( ) b a Inductance of length l: L = Φ B i = µ ( ) 0l b 2π ln a

20 Mutual Inductance An inductor can have an induced emf from its own changing magnetic field. It also can have an emf from an external changing field. That external changing field could be another inductor.

21 Mutual Inductance An inductor can have an induced emf from its own changing magnetic field. It also can have an emf from an external changing field. That external changing field could be another inductor. For self-inductance on a coil labeled 1: N 1 Φ B,1 = L 1 i 1 For mutual inductance: N 1 Φ B,2 1 = M 21 i 2 The flux is in coil 1, but the current that causes the flux is in coil 2.

22 Mutual Inductance For mutual inductance: N 1 Φ B,2 1 = M 21 i 2 UCTANCE The flux is in coil 1, but the current that causes the flux is in coil 2. N 1 N 1 Φ 12 B 1 N 2 Φ 21 B 2 N 2 B 2 i 1 i R + Coil 1 Coil 2 (a) + Coil 1 Coil 2 (b) R

23 Mutual Inductance mutual inductance M = N 1Φ B,2 1 i NDUCTION AND INDUCTANCE 2 = N 2Φ B,1 2 i 1 N 1 B 1 N 2 Φ 21 N 1 Φ 12 B 2 N 2 B 1 B 2 n. (a) The current i 1 in f i 1 is varied mf is induced n the meter les of the R i 1 i Coil 1 Coil 2 Coil 1 Coil 2 (a) (b) R

24 Mutual Inductance N 1 Φ B,2 1 = M 21 i 2 Considering the rate of change of both sides with time, and using Faraday s Law E = dφ B dt, and E 1 = M di 2 dt E 2 = M di 1 dt A change of current in one coil causes a magnetic flux in the other.

25 Summary energy stored in the magnetic field coaxial inductor mutual inductance applications of mutual inductance 4th Collected Homework due Thursday Mar 22. Quiz this Friday. Homework Serway & Jewett: Ch 32, onward from page 988. Obj. Qs: 1; Conc. Qs.: 7; Probs: 11, 15, 19, 33, 41, 43