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• Electricity and MagnetismEnergy of the Magnetic Field

Mutual Inductance

Lana Sheridan

De Anza College

Mar 14, 2018

• Last time

inductors

resistor-inductor circuits

• Overview

wrap up resistor-inductor circuits

energy stored in an inductor

coaxial inductor

mutual inductance

• Reminder: InductorsA capacitor is a device that stores an electric field as a componentof a circuit.

inductor

a device that stores a magnetic field in a circuit

It is typically a coil of wire.

• Reminder: Inductance

inductance

the constant of proportionality relating the magnetic flux linkage(NB) to the current:

NB = L I ; L =NBI

B is the magnetic flux through the coil, and I is the current inthe coil.

• RL Circuits: current rising

Current in loop:

974 Chapter 32 Inductance

After switch S1 is thrown closed at t ! 0, the current increases toward its maximum value e/R.

t !

L

t

R R0.632

i

e

t

Re

Figure 32.3 Plot of the current versus time for the RL circuit shown in Figure 32.2. The time constant t is the time interval required for i to reach 63.2% of its maximum value.

becomes zero and there is no time dependence of the current in this case; the cur-rent increases instantaneously to its final equilibrium value in the absence of the inductance. We can also write this expression as

i 5eR11 2 e2t/t 2 (32.7)

where the constant t is the time constant of the RL circuit:

t 5LR

(32.8)

Physically, t is the time interval required for the current in the circuit to reach (1 2 e21) 5 0.632 5 63.2% of its final value e/R. The time constant is a useful parameter for comparing the time responses of various circuits. Figure 32.3 shows a graph of the current versus time in the RL circuit. Notice that the equilibrium value of the current, which occurs as t approaches infinity, is e/R. That can be seen by setting di/dt equal to zero in Equation 32.6 and solving for the current i. (At equilibrium, the change in the current is zero.) Therefore, the current initially increases very rapidly and then gradually approaches the equilib-rium value e/R as t approaches infinity. Lets also investigate the time rate of change of the current. Taking the first time derivative of Equation 32.7 gives

didt

5eL

e2 t/t (32.9)

This result shows that the time rate of change of the current is a maximum (equal to e/L) at t 5 0 and falls off exponentially to zero as t approaches infinity (Fig. 32.4). Now consider the RL circuit in Figure 32.2 again. Suppose switch S2 has been set at position a long enough (and switch S1 remains closed) to allow the current to reach its equilibrium value e/R. In this situation, the circuit is described by the outer loop in Figure 32.2. If S2 is thrown from a to b, the circuit is now described by only the right-hand loop in Figure 32.2. Therefore, the battery has been eliminated from the circuit. Setting e 5 0 in Equation 32.6 gives

iR 1 L didt

5 0

The time rate of change of current is a maximum at t ! 0, which is the instant at which switch S1 is thrown closed.

didt

t

Le

Figure 32.4 Plot of di/dt versus time for the RL circuit shown in Fig-ure 32.2. The rate decreases exponen-tially with time as i increases toward its maximum value.

i(t) =E

R

(1 eRt/L

)

Derivative of current:

974 Chapter 32 Inductance

After switch S1 is thrown closed at t ! 0, the current increases toward its maximum value e/R.

t !

L

t

R R0.632

i

e

t

Re

Figure 32.3 Plot of the current versus time for the RL circuit shown in Figure 32.2. The time constant t is the time interval required for i to reach 63.2% of its maximum value.

becomes zero and there is no time dependence of the current in this case; the cur-rent increases instantaneously to its final equilibrium value in the absence of the inductance. We can also write this expression as

i 5eR11 2 e2t/t 2 (32.7)

where the constant t is the time constant of the RL circuit:

t 5LR

(32.8)

Physically, t is the time interval required for the current in the circuit to reach (1 2 e21) 5 0.632 5 63.2% of its final value e/R. The time constant is a useful parameter for comparing the time responses of various circuits. Figure 32.3 shows a graph of the current versus time in the RL circuit. Notice that the equilibrium value of the current, which occurs as t approaches infinity, is e/R. That can be seen by setting di/dt equal to zero in Equation 32.6 and solving for the current i. (At equilibrium, the change in the current is zero.) Therefore, the current initially increases very rapidly and then gradually approaches the equilib-rium value e/R as t approaches infinity. Lets also investigate the time rate of change of the current. Taking the first time derivative of Equation 32.7 gives

didt

5eL

e2 t/t (32.9)

This result shows that the time rate of change of the current is a maximum (equal to e/L) at t 5 0 and falls off exponentially to zero as t approaches infinity (Fig. 32.4). Now consider the RL circuit in Figure 32.2 again. Suppose switch S2 has been set at position a long enough (and switch S1 remains closed) to allow the current to reach its equilibrium value e/R. In this situation, the circuit is described by the outer loop in Figure 32.2. If S2 is thrown from a to b, the circuit is now described by only the right-hand loop in Figure 32.2. Therefore, the battery has been eliminated from the circuit. Setting e 5 0 in Equation 32.6 gives

iR 1 L didt

5 0

The time rate of change of current is a maximum at t ! 0, which is the instant at which switch S1 is thrown closed.

didt

t

Le

Figure 32.4 Plot of di/dt versus time for the RL circuit shown in Fig-ure 32.2. The rate decreases exponen-tially with time as i increases toward its maximum value.

di

dt=

E

LeRt/L

• RL Circuits: current rising

Potential drop across resistor:

80930-9 R L CI RCU ITSPART 3

VL (! L di/dt) across the inductor vary with time for particular values of !, L,and R. Compare this figure carefully with the corresponding figure for an RCcircuit (Fig. 27-16).

To show that the quantity tL (! L/R) has the dimension of time, we convertfrom henries per ohm as follows:

The first quantity in parentheses is a conversion factor based on Eq. 30-35, andthe second one is a conversion factor based on the relation V ! iR.

The physical significance of the time constant follows from Eq. 30-41. If weput t ! tL ! L/R in this equation, it reduces to

(30-43)

Thus, the time constant tL is the time it takes the current in the circuit to reachabout 63% of its final equilibrium value !/R. Since the potential difference VRacross the resistor is proportional to the current i, a graph of the increasingcurrent versus time has the same shape as that of VR in Fig. 30-17a.

If the switch S in Fig. 30-15 is closed on a long enough for the equilibriumcurrent !/R to be established and then is thrown to b, the effect will be to removethe battery from the circuit. (The connection to b must actually be made aninstant before the connection to a is broken. A switch that does this is called amake-before-break switch.) With the battery gone, the current through the resis-tor will decrease. However, it cannot drop immediately to zero but must decay tozero over time. The differential equation that governs the decay can be found byputting ! ! 0 in Eq. 30-39:

(30-44)

By analogy with Eqs. 27-38 and 27-39, the solution of this differential equationthat satisfies the initial condition i(0) ! i0 ! !/R is

(decay of current). (30-45)

We see that both current rise (Eq. 30-41) and current decay (Eq. 30-45) in an RLcircuit are governed by the same inductive time constant, tL.

We have used i0 in Eq. 30-45 to represent the current at time t ! 0. In ourcase that happened to be !/R, but it could be any other initial value.

i !!

R e"t/#L ! i0e"t/#L

L didt

\$ iR ! 0.

i !!

R (1 " e"1) ! 0.63

!

R.

1 H%

! 1 H%

! 1 V & s1 H &A " ! 1 %&A1 V " ! 1 s.

Fig. 30-17 The variation with time of(a) VR, the potential difference across theresistor in the circuit of Fig. 30-16, and (b)VL, the potential difference across the in-ductor in that circuit.The small trianglesrepresent successive intervals of one induc-tive time constant tL ! L/R.The figure isplotted for R ! 2000 %, L ! 4.0 H, and ! ! 10 V.

10 8 6 4 2

0 2 4 6 8

V R (

V)

t (ms)

0 2 4 6 8

V L (

V)

t (ms)

(a)

(b)

10 8 6 4 2

The resistor's potentialdifference turns on.The inductor's potentialdifference turns off.

CHECKPOINT 6

The figure shows three circuits with identical batteries, inductors, and resistors. Rankthe circuits according to the current through the battery (a) just after the switch isclosed and (b) a long time later, greatest first. (If you have trouble here, work throughthe next sample problem and then try again.)

(1) (2) (3)

halliday_c30_791-825hr.qxd 11-12-2009 12:19 Page 809

VR = iR

emf across inductor:

80930-9 R L CI RCU ITSPART 3

VL (! L di/dt) across the inductor vary with time for particular values of !, L,and R. Compare this figure carefully with the corresponding figure for an RCcircuit (Fig. 27-16).

To show that the quantity tL (! L/R) has the dimension of time, we convertfrom henries per ohm as follows:

The first quantity in parentheses is a conversion f

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