Physics 227: Exam 2 Information!
•
•
•
•
•
•
Exam 2: 16 questions covering chapters 25 – 28!
Thursday, Nov 17, 2011, 9:40 PM - 11:00 PM!
Room assignments:!
•
•
•
•
A-I ARC 103!
J-M SEC 111 (may start late)!
N-R PLH!
S-Z Beck Auditorium, Livingston Campus!!! (NOT Hill 114)!
Anyone with a conflict should contact Prof. Cizewski
cizewski@rutgers.edu TODAY!!!
Bring pencils, one formula sheet w/ anything you want !
NO calculators, NO cell phones, NO electronics needed or
allowed!!
Outline Lecture 20
•  Review of Lecture 19 & Faraday’s law
•  Self inductance and inductors
•  How a time-varying current in one coil can
induce an emf
•  How to relate the induced emf in a circuit to
the rate of change of current in the same
circuit
•  The R-L circuit:
how to analyze circuits that
include both a resistor and an inductor (coil)
•  Magnetic field energy – how to calculate the
energy stored in a magnetic field
Lecture 19 Review!
•  Maxwell’s equations
•  Gauss’s Law for E
•  Gauss’s Law for B
∫
qenclosed
E• d A =
ε0
→
→
→
→
∫ B• d A = 0
(no magnetic charges)
€
•  Ampere’s Law
(conduction + displacement currents)
€
dΦE
1
∫ B• d  = µ0 (iconduction + ε0 dt )enclosed where ε µ = c
0 0
→
→
•  Faraday’s Law
€
∫
dΦB
E• d  = −
dt
→
→
Physics 227: Lecture 20!
•  Faraday’s Law:!
dΦB
EMF = −
where
dt
→
ΦB =
→
∫ B• d A = BA cosθ (if B is uniform over A)
•  If change in magnetic flux, then EMF is induced!
Note minus sign (Lenz’s Law):
self-induced EMF opposes change in current
Faraday’s Law PhET
Isolated circuit !
and induced EMF!
Induced
EMF!
•  Previous assumption not quite correct:
When switch closed I≠ε/R immediately!
•  Faraday’s law prevents from happening!
•  Rather, as current increases vs time,
Isolated circuit: switch,
resistor, source of EMF!
ΦΒ due to this current also increases!
•  This increasing flux induces EMF that
opposes change in ΦΒ => opposing EMF
means gradual increase in the current!
•  This effect = SELF-INDUCTION!
€
dΦB
d
EMF = −
= − BA
dt
dt
Isolated circuit !
and induced EMF!
•  When switch closed I≠ε/R immediately!
•  Faraday’s law prevents from happening!
•  Rather, as current increases vs time,
ΦΒ due to this current also increases!
•  This effect = SELF-INDUCTION = L!
Isolated circuit: switch,
resistor, source of EMF!
dΦB
di
EMFinduced = −N
= −L
dt
dt
NΦB
L=
i
Assume same flux through all N turns!
Inductance depends upon geometry!
Inductors are circuit elements!
•  Purpose of inductor:
oppose any
variations in the current through
the circuit; !
•  Unit of inductance = !
#Henry = Volt-second/Ampere!
•  Examples:!
•  Inductor in a DC circuit helps to
maintain a steady current
despite fluctuations in the
applied EMF!
•  Inductor in an AC circuit tends
to suppress variations of the
current that are more rapid
than desired!
Circuits with EMF and Inductor!
Resistance = !
#measure of opposition to current!
  Voltage drop across resistance depends on current!
Inductance = !
#measure of opposition to change in current!
  Voltage drop across inductor depends on change in
current, di/dt!
Circuits with EMF and Inductor!
a. Voltage drop across resistor:
Vab==Va-Vb=iR > 0
b. Inductor:
constant current i; no
potential difference!
#Vab=L•di/dt=0
c. Inductor with increasing current I!
#Vab=L•di/dt>0!
d. Inductor with decreasing current I !
#Vab=L•di/dt<0!
I-clicker
A current i flows through an
inductor L in the direction from
point b toward point a. There is
zero resistance in the wires of the
inductor. If the current is
decreasing,
A. the potential is greater at point a than at point b.
B. the potential is less at point a than at point b.
C. the answer depends on the magnitude of di/dt
compared to the magnitude of i.
D. The answer depends on the value of the inductance
L.
E. both C. and D. are correct.
I-clicker
A current i flows through an
inductor L in the direction from
point b toward point a. There is
zero resistance in the wires of the
inductor. If the current is
decreasing,
⇐ EMF
A. the potential is greater at point a than at point b.
€
B. the potential is less at point a than at point b.
C. the answer depends on the magnitude of di/dt
compared to the magnitude of i.
D. The answer depends on the value of the inductance
L.
di
EMFinduced = Va − Vb = −L > 0
E. both C. and D. are correct.
dt
Example: Inductance of Solenoid!
•  Given:
uniformly wound, air-filled solenoid with !
#N turns and length , where  >> radius r!
•  What is the inductance of the solenoid?!
N
B = µ0 nI = µ0 I

•  First calculate B of solenoid!
•  Then calculate flux through each turn!
N
ΦB = BA = µ0 IA

•  From definition of €inductance: !
NΦB N
NIA
N 2A
(n) 2 A
L=
= (µ0
) = µ0
= µ0
= µ0 n 2 A
I
I



€
•  Inductance of solenoid proportional to volume of solenoid!
Solenoid image: hyperphysics.phy-astr.gsu.edu
Example: Inductance of toroid!
  Given toroidal solenoid!
  A=cross sectional area!
  R=mean radius!
  N turns!
  Non-magnetic core!
  What is self-inductance?!
  Need Φ first, calculate B!
B
  (B≠0 when inside of windings) !
  Calculate L!
2
NΦB µ0 N A
L≡
=
i
2πr
ΦB = BA = µ0 niA
N
n=
2πr
µ0 NiA
ΦB =
2πr
Example: Inductance of toroid!
  Given toroidal solenoid!
  A=cross sectional area!
  R=mean radius!
  N turns!
  Non-magnetic core!
  What is self-inductance?!
  What is magnitude and
2
NΦB µ0 N A
L≡
=
i
2πr
direction of induced EMF
when di/dt>0?!
2
di
µ0 N A di
EMF = −L = −(
)
dt € 2πr dt
Opposes EMF that
gives rise to i!
R-L Circuits!
Circuit: resistor, inductor, battery!
Close switch at t=0!
Current increases!
How does I increase vs time?!



  Inductor produces back emf =
opposes increasing current!
€
dI
εL = −L
dt
  Apply Kirchhoff’s loop equation! ε − IR − L dI = 0
€
dt
€
R-L Circuits!
Circuit: resistor, inductor, battery!
How does the current increase as
a function of time? Solve for I!

ε
L dI
−I−
=0
R
R dt
  Choose variable x so that!
ε
x = − I and dx = −dI
R
  Then loop equation:!
dx
R
= − dt
x
L
€
x
R
integrating : ln = − t
x0
L
taking antilog :
−Rt / L
€
−t / τ
x = x 0e
= x 0e
where τ = L /R
R-L Circuits!
Circuit: resistor, inductor, battery!
How does the current increase as
a function of time?!
Solved for! x = x 0e−Rt / L = x 0e−t / τ


where τ = L /R
at t = 0, I = 0
  Recall variable!
€
€
ε
x = − I ⇒ x 0 (t = 0) = ε /R
R
ε
−t / τ
  Current vs time: ! I = (1− e )
€
R
€
dI
ε − IR − L = 0
dt
R-L Circuits!
Circuit: resistor, inductor, battery!
Current increase as a function of
time?!
−t / τ

ε
I = (1− e )
R
τ = L /R
€
  Final current does not depend on
€ 
L, I=ε/R as if no inductor in
circuit!
Rate of increase of I, dI/dt, is
maximum at t=0 and falls of
exponentially as t=>∞
dI/dt vs t
I clicker: time constant of R-L circuit
An inductance L and a
resistance R are connected to
a source of emf as shown.
When switch S1 is closed, a
current begins to flow. The
time required for the current
to reach one-half its final
value is
A. directly proportional to RL.
B. directly proportional to R/L.
C. directly proportional to L/R.
D. directly proportional to 1/(RL).
E. independent of L.
I clicker: time constant of R-L circuit
An inductance L and a
resistance R are connected to
a source of emf as shown.
When switch S1 is closed, a
current begins to flow. The
time required for the current
to reach one-half its final
value is
A. directly proportional to RL.
B. directly proportional to R/L.
C. directly proportional to L/R.
D. directly proportional to 1/(RL).
E. independent of L.
I=
I
ε
ε
(1− e−t / τ ) = f =
R
2 2R
(1− e−t / τ ) = 1/2
e−t / τ = 1−1/2 = 1/2
et /τ = 2
t / τ = ln2
t = τ (ln2) but τ = L /R
t proportional to τ = L /R
Time constants τ are short in R-L Circuits!
Circuit: resistor, inductor, battery!
Time constant τ=L/R!
L= 1 Henry, R=100 Ohms!
τ = 10-2 seconds or 10 ms!



#I clicker example: time to get to
½ Imax !
t= τln(2)!
τ=10-2 s => t=6.9 ms!


€
  Inductor produces back emf =
opposes increasing current!
Recall Lecture 7!
Capacitors, Capacitors, Capacitors!
•  Defined capacitors and capacitance: Q=CV!
•  Calculated energy stored in capacitor!
1 Q2 1
U=
= CV 2
2 C
2
•  Energy density stored in capacitor!
1
2
u = ε0 E
2
€
•  What about inductors and energy storage?!
€
Energy in magnetic field!
  Back to example of
increasing current!
#Vab=Va-Vb=L di/dt>0!
But power!

P = Vab i
di
P = Li
dt
  Energy supplied in dt:!
di
dU = Pdt = (L )idt
dt
€
€
dU = Lidi
1 2
U = L ∫ 0 idi = LI
2
I
Inductor is a device
in which energy is
stored
I-clicker
A steady current flows through an inductor. If the
current is doubled while the inductance remains
constant, the amount of energy stored in the
inductor
A. increases by a factor of √2 .
B. increases by a factor of 2.
C. increases by a factor of 4.
D. increases by a factor that
depends on the geometry of the
inductor.
E. none of the above
I-clicker
A steady current flows through an inductor. If the
current is doubled while the inductance remains
constant, the amount of energy stored in the
inductor
A. increases by a factor of √2 .
B. increases by a factor of 2.
C. increases by a factor of 4.
D. increases by a factor that
depends on the geometry of the
inductor.
E. none of the above
1 2
U i = LIi
2
1 2 1
2
U f = LI f = L(2Ii ) = 4U i
2
2
Magnetic Energy Density!
  Recall:
energy density stored in a capacitor,
energy stored in E field!
U
1
2
u=
= ε0 E
volume 2
€
€

Magnetic Energy Density in a
Solenoid?!
B = µ0 ni
Earlier result for solenoid!
B
⇒i=
µ0 n
L = µ0 n 2 A
  Energy stored 1in !
#solenoid!
2
1
B
1
B
UL = i 2L = (
) 2 (µ0 n 2 A) =
(A)
2
2 µ0 n
2 µ0
€
1 B2 €
UL =
(volume of solenoid)
2 µ0
UL
1 B2
uL =
=
in vacuum
volume 2 µ0
  Energy density stored in
€
solenoid!
1 B2
uL =
in material
2µ
Demo: L-R circuit!
  Turn on circuit:
light bulb
lights up immediately!
Magnetic field in solenoid
is small!
What happens if insert the
following into the solenoid?!
Wood, copper pipe, iron
pipe, bundle of iron fibers?!



€
  B depends on µ=K µ
  Energy stored depends on
m 0!
χm=Km-1 (see Table 28.1)!
€
UL
1 B2
uL =
=
in vacuum
volume 2 µ0
1 B2
uL =
in material
2µ
Electrical Energy Density stored in a Capacitor
vs !
Magnetic Energy Density stored in a Solenoid!
  Electrical Energy
density stored in a
capacitor from energy
stored in E field!
UE
1
uE =
= ε0 E 2 in vacuum
volume 2
1 2
uE = εE with dielectric
2
€

2
U
1
B
€
L
u
=
=
in vacuum
L
Magnetic Energy density
volume 2 µ0
stored in a solenoid from
1 B2
in material
energy stored in B field! uL =
2µ
Summary Lecture 20
•
Self inductance and inductors
•
•
•
•
How a time-varying current in one coil
can induce an emf
dΦB
di
EMFinduced = −N
= −L
dt
dt
How to relate the induced emf in a
circuit to the rate of change of current L = NΦ B
in the same circuit
i
The R-L circuit: how to analyze circuits
that include both a resistor and an
inductor (coil)
€
Magnetic field energy –
how to calculate the energy
stored in a magnetic field
ε
I = (1− e−t / τ )
R
τ = L /R
UL
1 B2
uL =
=
in vacuum
volume 2 µ0
€
1 B2
uL =
in material
2µ
Thank you!!
On Thursday Nov 17: Review of Chapters
25-28!
Reminder Physics 227 Exam 2 !
•
•
•
•
•
Thursday Nov 17 9:40-11:00 PM!
Exam 2: 16 questions covering chapters 25 – 28!
Room assignments:!
A-I ARC 103!
J-M SEC 111 (may start late)!
N-R PLH!
S-Z Beck Auditorium, Livingston Campus!!! (NOT Hill 114)!
Anyone with a conflict (≥2 exams) should contact Prof. Cizewski
cizewski@rutgers.edu TODAY!!!
Bring pencils, one formula sheet w/ anything you want !
NO calculators, NO cell phones, NO electronics needed or allowed!!
•
•
•
•