PS 250: Lecture 26 Energy Storage and

RL CircuitsJ. B. Snively

November 4th, 2015

Today’s Class

Magnetic Field EnergyR-L CircuitsSummary

Inductors

E

E = �Ldi

dt

di

dt L

a

b

Recall that:

Therefore: Vab = Ldi

dt

As a circuit element, electromotive force opposes current

changes in an inductor!

Magnetic Field EnergyAnalogous to a capacitor’s storage of energy in an electric field, inductors

store energy in a magnetic field.

P = Vabi = Lidi

dt

Rate of energy delivered to the inductor is:

Recall that: P =dU

dtSo, dU = Li di

U = L

Z I

0idi =

1

2LI2

Magnetic Energy Density

u =B2

2µo

Energy per unit volume (u=U/V):

In a material where µ≠µo:

u =B2

2µµ = K

m

µo

(“Magnetic materials”, for example, may have very high permeability µ >> µo)

Today’s Class

Magnetic Field EnergyR-L CircuitsSummary

Initial current i=0, di/dt=E/L

Initial potential difference = vbc = EInductor behaves initially like an open circuit!

R-L Circuit at Turn-On At t=0...

E+

-

vab=0

vbc=ELR

t=0

Final current I=E/R

Final potential difference = vbc = 0 Inductor eventually behaves like a short circuit! (Note: real inductors have resistance!)

R-L Circuit at Steady-State At t=∞...

E+

-

vab=E

vbc=0LR

t=∞

R-L Circuit Current Growth First switch closed, second switch still open.

E+

- LRt=0

i(t) =ER(1� e�(R/L)t)

di

dt=

ELe�(R/L)t

R-L Circuit Current Decay First switch opened, second switch closed at

the same time!

E+

- LRt=0

i(t) =ERe�(R/L)t

i(t) = Io

e�(R/L)tFor an initial current Io

Summary / Next Class:

Mastering Physics for Friday

Homework for Friday