Energy stored in a magnetic field Energy Stored in an Inductor Energy - - PowerPoint PPT Presentation

Energy stored in a magnetic field

Energy Stored in an Inductor

2

LI 2 1 U  dt dI L

• 



Energy stored in an inductor:

L

(Do not forget .) Energy density stored in an electric field:

2 B B

B 2 1 U u    

Capacitor and Inductor Capacitor C Inductor L Charge Q Current I E field B field

Parallel plate capacitor (uniform E field) Solenoid (uniform B field)

C Q V  t d I d L

• 



d V E and d A C    nI B and nNA L    

2 E 2 E

E 2 1 u and CV 2 1 U   

2 B 2 B

B 2 1 u and LI 2 1 U   

Class 40 RL Circuits

RC Circuits – Charging

 C R

At t=0, capacitance is uncharged and Q=0 (initial condition). At t=0, switched is closed, if the capacitor has no charge, it behaves like a conductor and I=/R. After the capacitor is completely charged, Q=C , VC=  and VR=0. I=0 and the capacitors behave like an insulator. From Class 25 Charge

RL Circuits – Charging

 L R

At t=0, inductor is uncharged and I=0 (initial condition). At t=0, switched is closed, if the inductor has no current, it behaves like an insulator (opposes changes) and I=0. After the inductor is completely charged (with current), I=/R, VL= 0 and VR= . The inductor behaves like a conductor. Current

RC Circuits – Charging

 C R

) e 1 ( C q C

• K

K C q 0, At t e K C q ) e (K Ke C

• q

K' CR t

• )

C

• q

n( dt CR 1

• C
• q

dq dt q)

• (C

dq CR t d q d R C q IR C q

CR t

• CR

t

• K'

CR t

• 

                                      ) e

• (1

C q V e IR V e R e CR C t d dq I

CR t

• C

CR t

• R

CR t

• CR

t

• 

           

Integration constant VR + VC = 

From Class 25 Charge

RL Circuits – Charging

) e 1 ( R I ) e 1 ( R I K K I 0, At t Ke IR ) e (K Ke IR

• K'

L R t L R ) IR

• n(

K' t ) IR

• n(

R L dt IR

• dI

L IR)dt

• (

dI L dt dt R I dI L IR t d I d L

t L R

• t

L R

• t

L R

• K'

L R t L R

• 

                                              

t L R

• e

dt dI L V ) e 1 ( IR V

L t L R

• R

        

Integration constant VR + VC = 

Current

 L R

RC time constant

=RC is known as the RC time constant. It indicates the response time (how fast you can charge up the capacitor) of the RC circuit.

e R I

CR t

• 

 R I  

t

R 37 . ~ R e I

1

• 

 

t=RC

) e 1 ( C q

CR t

• 

   C q 

t

  C 63 . ~ C ) e 1 ( q

• 1

 

t=RC

37 . e 2.72 e

1

• 

 707 . 2 1 1.414 2  

Nothing to do with RC circuits

From Class 25

L/R time constant

=L/R is known as the time constant. It indicates the response time (how fast you can up a current) of the RC circuit.

e L t d I d

t L R

• 

 L dt dI  

t

L 37 . ~ L e dt dI

1

• 

 

t=L/R

) e 1 ( R I

t L R

• 

  R I  

t

R 63 . ~ R ) e 1 ( I

1

• 

  

t=L/R

37 . e 2.72 e

1

• 

 707 . 2 1 1.414 2  

Nothing to do with RL circuits

RC Circuits – Discharging

CR t

• CR

t

• K'

CR t

• Qe

q K Q Q q 0, At t e K q ) e (K Ke q K' CR t

• q

n dt CR 1

• q

dq dt q

• dq

CR t d q d R C q IR C q                        

CR t

• C

CR t

• R

CR t

• e

C Q C q V e C Q IR V e RC Q t d dq I          

Integration constant VR + VC = 0

C R

Charge From Class 25