Electricity and Magnetism Inductance

Transcription

1 Electricity and Magnetism Inductance ana Sheridan De Anza College Mar 13, 2018

2 ast time relativity and fields

3 Overview inductors and inductance resistor-inductor circuits

4 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 Circuit substitution problem. of is superconducting magne Notice that this the as Equation 26.2 Notice that thisexpression expression is the samesame asthis Equation 26.2, the capa Capacitor section. Th approximately ten times g component symbols symbol initially unchar Use Equation to express the magnetic field in the tromagnets. Such superco In Supercond studying e storing energy. interior of the base solenoid: 26.3circuit Combinations resonance imaging, or MR diagram " Battery 26.3 Combin Find the mutual inductance, noting that magnetic Twothe or more capacitors oftenfo a organs without the need elements. The symbol battery V the equivalent capacitance of c! coil caused flux F BH through the handle s by the magful radiation. orthroughout more capacito Capacitor this Two section. this se wires between symbol initially theuncharged. equivalent capac netic field the base coil is BA: The of direction of the effective flowcapacitor of positive and switches asw In studying electric circuits, this section. Through diagram. Such The a diagra "a number circuit Switch Wireless charging is used ofinitially other cordless ds Battery in ure Open charge is clockwise. capacitor Csymbol 27.6 Electrical elements. Theuncharged. circuit symbolsp symbol! symbol used by some manufacturers of electricwires cars that avoids model for a dire cap In studying electric between the circuit elem I andcircuit switches as wellcircuits, as thesuch colo diagram. In typical electric ing apparatus. Closed is at the higher " Battery Switch ure The symbol for the ca Open elements. The circuit a source such as a battery symbol model for a capacitor, a pair of! switch Figure 26.6 Ssymbol CircuitClosed symbols forisdetermine between the atwires the higher potential and ci is r et s an express batteries, and switches. Parallel Com and switches as well b capacitors, c symbols Figure 26.6 Circuit for transfer. First, consider thea capacitors, batteries, and Parallel Combination Notice that capacitors areswitches. in Switch uretwo The symbol # Open capacitors to a resistor. (Resistors are Notice that capacitors are in Rin green, Two capacitors as sho!v batteries blue, are and symbol model forconnected a capacitor blue, resistor R batteries are in green, andconnecting " wires also have nation of capacitors. Figure 26. nation of capac switches are in red. The closed switches are in red. The closed Closed is at the higher poten a d whereas capacitors. The left plates of the switch can carry current, some to the resistor. Unles capacitors. The switch can whereas the battery by a conducting wire thecarry open onecurrent, cannot.! Figure 26.6 Circuit symbols for ais capacitor When is conn wires small the open one batteries, cannot. the compared battery bywit a C capacitors, and switches. Parallel Combinat " inductor delivered to the wires is ne combination is an C circu Q max Notice that capacitors are in Two capacitors following acurr pos blue, batteries are in green,then and Imagine closed, both theconne Figure A circuit consistnation of capacitors. switches are in red. The closedcircuit in Figure from late between maximum p ing of a resistor of can resistance R capacitors. The left pl switch carry current, whereas We identify the entire circu S and a battery having a potential cuit is zero, no energy is tr the open one cannot Oscillations the battery by a condu

6 Inductance Just like capacitors have a capacitance that depends on the geometry of the capacitor, inductors have an inductance that depends on their geometry. Capacitance is defined as being the constant of proportionality relating the charge on the plates to the potential difference across the plates Q = C ( V ). Inductance is defined in a similar way.

7 Inductance The magnetic flux linkage is NΦ B. Inductance, the constant of proportionality relating the magnetic flux linkage (NΦ B ) to the current, I: NΦ B = I ; = NΦ B I Φ B is the magnetic flux through the coil, and I is the current in the coil. Units: henries, H. 1 henry = 1 H = 1 T m 2 / A

8 Inductance Just like capacitors have a capacitance that depends on the geometry of the capacitor, inductors have an inductance that depends on their structure. For a solenoid inductor: = µ 0 n 2 Al where n is the number of turns per unit length, A is the cross sectional area, and l is the length of the inductor. (Doesn t depend on current or flux, only geometry of device.)

9 Value of µ 0 : New units The magnetic permeability of free space µ 0 is a constant. µ 0 = 4π 10 7 T m / A It can also be written in terms of henries: µ 0 = 4π 10 7 H / m (Remember, 1 H = 1 T m 2 / A)

10 Inductance of Solenoid Inductors Suppose now that the only source of magnetic flux in the solenoid is the flux produced by a current in the wire. Then the field produced within the solenoid is: B = µ 0 In where n is the number of turns per unit length. That means the flux will be: Φ B = BA cos(0 ) = BA = µ 0 InA where A is the cross sectional area of the solenoid.

11 Inductance of Solenoid Inductors = NΦ B I Replacing N = nl, Φ B = µ 0 InA: = nl(µ 0In)A I So we confirm our expression for a solenoid inductor: = µ 0 n 2 Al

12 Induction from an external flux vs Self-Induction Previously, we considered the effect of a changing magnetic field from some external source causing and emf and current flow in a wire loop. However, a changing current in the solenoid itself can be causing the changing field that affects the flow of the current. 1 In this textbook, and most sources, is the self-inductance.

13 Induction from an external flux vs Self-Induction Previously, we considered the effect of a changing magnetic field from some external source causing and emf and current flow in a wire loop. However, a changing current in the solenoid itself can be causing the changing field that affects the flow of the current. This inductance 1,, is the self-inductance of the coil. NΦ B = I We are assuming the flux Φ B is entirely due to the B-field resulting from the current in the solenoid and there is no external source of magnetic field adding to the flux. 1 In this textbook, and most sources, is the self-inductance.

14 Self-Induction When the current in the solenoid circuit is changing there is a (self-) induced emf in the coil. From Faraday s aw, we have E = d(nφ B) dt Since is a constant for a particular inductor and i = NΦ B, E = di dt E is the self-induced emf. The emf opposes the change in current.

15 Inductors vs. Resistors Inductors are a bit similar to resistors. Resistors resist the flow of current. Inductors resist any change in current. If the current is high and lowered, the emf acts to keep the current flowing. If the current is low and increased, the emf acts to resist the increase.

16 (32.3) magnetic flux changes in time Self-Induction and induces an emf in the loop. by Equaematical tion to a quation S S B i try. This geomedepenductance, but the aluated. R i Figure 32.1 Self-induction in a simple circuit. e

17 Self-Induction 30-9 R CIRCUITS 807 PART 3-13, flux that arge s an n of agthe ging emf real i (increasing) (a) i (decreasing) The changing current changes the flux, which creates an emf that opposes the change. i (decreasing) (b) c th th

18 nless otherwise indicated, we assume here Self-inductance question The figure shows an emf E induced in a coil. a coil. Which of ough the coil: (a) leftward, (c) inand rightward, and leftward? Which of the following can describe the current through the coil: (A) constant and rightward (B) increasing and rightward (C) decreasing and rightward (D) decreasing and leftward enly introduce an emf into a single-loop (b) Fig and the self-in the coil in a dir the increase.t be drawn alon side the coil. B rent i is decrea appears in a di the decrease.

19 nless otherwise indicated, we assume here Self-inductance question The figure shows an emf E induced in a coil. a coil. Which of ough the coil: (a) leftward, (c) inand rightward, and leftward? Which of the following can describe the current through the coil: (A) constant and rightward (B) increasing and rightward (C) decreasing and rightward (D) decreasing and leftward enly introduce an emf into a single-loop (b) Fig and the self-in the coil in a dir the increase.t be drawn alon side the coil. B rent i is decrea appears in a di the decrease.

20 R Circuits introduce tor R and e,the cure current er,a selfthe rise of Thus, the onstant s long as Just like circuits with capacitors and resistor, circuits with inductors and resistors have time-dependent behavior. less rapid to di/dt, it. ptotically. + a b S Fig An R circuit. When switch Initially, an inductor acts to oppose changes in the current through S is closed on a,the current rises and approaches a limiting value /R. A long time later, the current stabilizes and it acts like ordinary connecting wire. R

21 owever, that even a circuit and an emf that opposes the cuit s behavior. increasing current is induced k emf, R an Circuits inductor in Question a cirinductor attempts to keep in the inductor. d. If the battery voltage in a S R 2 uctor opposes this change b is decreased, the inductor S 1 iate drop. Therefore, the changes in the voltage. ains a battery of negligible e the elements connected to s on switch S 2 suggest this (If the switch is connected ps.) Suppose Consider S 2 the is set circuit to a shown When with the switch S S 2 is thrown 1 open and S 2 at position a. to position b, the battery is no t t 5 0. Switch The current S 1 is in now the thrownlonger closed. part of the circuit and at opposes the increasing the current decreases. At the instant it is closed, across which circuit element is the ule to voltage this circuit, equal travers- to the emf Figure of 32.2 the battery? An R circuit. When switch S 2 is in position a, (A) the resistor the battery is in the circuit. (B) the inductor (32.6) (C) both each of the inductor and the resistor off s rules were developed plied to (D) a circuit neither in which

22 owever, that even a circuit and an emf that opposes the cuit s behavior. increasing current is induced k emf, R an Circuits inductor in Question a cirinductor attempts to keep in the inductor. d. If the battery voltage in a S R 2 uctor opposes this change b is decreased, the inductor S 1 iate drop. Therefore, the changes in the voltage. ains a battery of negligible e the elements connected to s on switch S 2 suggest this (If the switch is connected ps.) Suppose Consider S 2 the is set circuit to a shown When with the switch S S 2 is thrown 1 open and S 2 at position a. to position b, the battery is no t t 5 0. Switch The current S 1 is in now the thrownlonger closed. part of the circuit and at opposes the increasing the current decreases. At the instant it is closed, across which circuit element is the ule to voltage this circuit, equal travers- to the emf Figure of 32.2 the battery? An R circuit. When switch S 2 is in position a, (A) the resistor the battery is in the circuit. (B) the inductor (32.6) (C) both each of the inductor and the resistor off s rules were developed plied to (D) a circuit neither in which

23 owever, that even a circuit and an emf that opposes the cuit s behavior. increasing current is induced k emf, R an Circuits inductor in Question a cirinductor attempts to keep in the inductor. d. If the battery voltage in a S R 2 uctor opposes this change b is decreased, the inductor S 1 iate drop. Therefore, the changes in the voltage. ains a battery of negligible e the elements connected to s on switch S 2 suggest this (If the switch is connected ps.) Suppose Consider S 2 the is set circuit to a shown When with the switch S S 2 is thrown 1 open and S 2 at position a. to position b, the battery is no t t 5 0. Switch The current S 1 is in now the thrownlonger closed. part of the circuit and at opposes the increasing the current decreases. After a very long time, across which circuit element is the voltage ule to equal this circuit, to thetravers- emf of the Figure battery? 32.2 An R circuit. When switch S 2 is in position a, (A) the resistor the battery is in the circuit. (B) the inductor (32.6) (C) both each of the inductor and the resistor off s rules were developed plied to (D) a circuit neither in which

24 owever, that even a circuit and an emf that opposes the cuit s behavior. increasing current is induced k emf, R an Circuits inductor in Question a cirinductor attempts to keep in the inductor. d. If the battery voltage in a S R 2 uctor opposes this change b is decreased, the inductor S 1 iate drop. Therefore, the changes in the voltage. ains a battery of negligible e the elements connected to s on switch S 2 suggest this (If the switch is connected ps.) Suppose Consider S 2 the is set circuit to a shown When with the switch S S 2 is thrown 1 open and S 2 at position a. to position b, the battery is no t t 5 0. Switch The current S 1 is in now the thrownlonger closed. part of the circuit and at opposes the increasing the current decreases. After a very long time, across which circuit element is the voltage ule to equal this circuit, to thetravers- emf of the Figure battery? 32.2 An R circuit. When switch S 2 is in position a, (A) the resistor the battery is in the circuit. (B) the inductor (32.6) (C) both each of the inductor and the resistor off s rules were developed plied to (D) a circuit neither in which

25 R Circuits oop rule: x + i R Fig The circuit of Fig with the switch closed on a.we apply the loop rule for the circuit clockwise, starting at x. E E ir = 0 y z Here t,the inductiv et s examine E and for a time long a Eq , the expon rent is initially i 0, tial goes to e 0. rium value of /R. We can also exam shows how th E di dt ir = 0 This is a differential equation. Solution: i = E R (1 e t/τ ) ; τ = R

26 Current varies with time E di dt ir = 0 Rearranging: di dt di dt 1 E/R i di = = E i R = R ( ) E R i R dt The limits of our integral will be determined by the initial conditions for the situation we are considering.

27 R Circuits: current rising When charging an initially uncharged capacitor: i = 0 at t = 0 i 0 1 E/R i di = t 0 R dt ln(e/r i) + ln(e/r 0) = R t ( ) E/R ln E/R i = Rt E/R E/R i = e Rt/ The solution is: (1 e Rt/) i(t) = E R

28 fter Rswitch 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 R t R e di dt e 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 R e Rt/) circuit R sus time for the R 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 R circuit 32.4 shown Plot of in di/dt Fig- versu dt = E e Rt/ ure time The for rate the decreases R 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.

29 or R 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 (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 (V) t (ms) (b) E (t) = di Fig The variation dt with time o (a) V R,the potential difference across th resistor in the circuit of Fig , and ( V,the potential difference across the in

30 R Circuits: Time Constant τ = /R τ is called the time constant of the circuit. (Notice that it is defined differently for R 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 R-circuits.

31 ccurs if we introduce ining Ra resistor Circuits R and 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 s becomes less rapid Fig An R 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. oop rule: E ir = 0 Solution: di dt ir = 0 i = E R e t/τ ; τ = R

32 At t 0, the switch is thrown to R 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 ; τ shown in Figure = R32.2. For t, 0, switch S 2 is at position a.

33 Summary inductance R circuits Homework Serway & Jewett: Ch 32, onward from page 988. Obj. Qs: 3, 5; Conc. Qs.: 5; Probs: 1, 3, 9