RL Circuit

0

0

iRdt

diL

iRVL

iR

didt

L

dt

diLiR

1

t=0, i=0 LRteR

i /1

R/L

Switch to position a

Switch to position bLRtLRt eie

Ri /

0/

Initially, i change is max, thus largest VL. After t>>all voltage is on R, di/dt=0, so VL =0

In a dc circuit, inductor behaves like a short circuit

Inductor & Capacitor in DC Circuit

If there is a sudden change in current or Voltage occurs in a circuit such as close or open a switch, then

Inductor Capacitor

Current (iL) must be continuous,

i.e. i+=i-

Voltage (Vc) must be continous,

i.e. V+=V-

At t>>

0dt

diLVL Short circuit 0

dt

dqiC Open circuit

Magnetic field energy stored in an inductor:

Ridt

diiLi

iRdt

diL

2

Power supplied by battery

Dissipated power

Work stored

2

2

2

1LiU

dt

dU

dt

diL

dt

diiLP

B

B

Concept Check

A battery is connected to a solenoid. When the switch is opened, the light bulb

1. Remain off2. Goes off3. Slowly dims out4. Keeps burning as brightly as it did before

the switch was opened.5. Flares up brightly, then dims and goes out

Answer 5

LC Circuita) Charged C connected L

Vmax=qmax/C, i = 0, di/dt: max

UE=qmax2/2C, max

UB=Li2/2=0b) U=UB+UE

c) imax, q=0, UB max

LC oscillationVmax=qmax/C, i=0

UE=qmax2/2C, max

UB=Li2/2=0

Speed of charging depends on L, C

UE=q2/2C, UB=Li2/2

q=0, imax

UE=q2/2C=0UB=Limax

2/2, max

Vmax=qmax/C, i=0

UE=qmax2/2C, max

UB=Li2/2=0

The charge starts to flow back the other way, resulting opposite current

LC oscillation

LC oscillation The oscillations continuous indefinitely in the absence of loss (R=0)

The Vc (or charges) is out of phase with i, i.e. Vc max. at i=0, vice versa.

LC circuit Oscillating block-spring systems

q Displacement: x

i=dq/dt v=dx/dt

L m

C 1/k

UB=Li2/2 Uk=mv2/2

UE=q2/2C U=kx2/2

mkLC

1

0d

m

energyconstant for 0/2

1

2

1

2

2

22

kxdt

x

dtdU

kxmvU

tXx cos

tQqC

q

dt

q

cos

0d

L2

2

LC oscillation Circuit

0/

22

1 22

dt

dq

C

q

dt

diLidtdU

C

qLiU

0 Q,q 0, tif

cos

0d

L2

2

tQq

C

q

dt

q

fTf

LC

C

tQtLQ

1,

2,

1

0cos

cos2

Concept CheckWhich Circuit takes the least time to fully discharge the capacitors during the oscillation

(a)

LCf

TfLC

2

1,

2,

1

Answer: (b) has smaller Ceq, thus smaller T, fast discharge

(b)

Example: RC circuit

33-19P, In an oscillating LC circuit, L=3.0 mH and C=2.60 F. At t=0 the charge on the capacitor is zero and the current is 2.00 A. (a) what is the maximum charge that will appear on the capacitor? (b) In terms of the period T of the oscillation, how much time will elapse after t=0 until the energy stored in the capacitor will be increasing at its greatest rate? c) What is this greatest rate at which energy is transferred to the capacitor?

sec41.14/

(sec)1065.5

LC2/2/1

1080.1

1070.21000.32

LC1 Q.i 0, t

cos

sin

4

4

63

Tt

fT

C

LCIQ

at

tQdt

dqi

tQq

tC

Q

C

ttQ

dt

dU

C

tQ

C

qU

E

E

2sin2

cossin

2

sin

2

2

2

222

Damped and Forced Oscillations

Damped Oscillation Forced Oscillation