![Page 1: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/1.jpg)
Self Inductance• Consider a solenoid L, connect it to a battery
• Area A, lengthl, N turns • What happens as you close the switch?• Lenz’s law – loop resists change in magnetic
field• Magnetic field is caused by the current• “Inductor” resists change in current
E +–
0NIB
0
1B
NIA
B
L
d
dt
E 1B
dN
dt
20N IAd
dt
20N A dI
dt
dI
Ldt
E 20L N A
A
l
Ch. 32
![Page 2: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/2.jpg)
Warmup 18
![Page 3: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/3.jpg)
Inductors• An inductor in a circuit is denoted by this
symbol:• An inductor satisfies the formula:• L is the inductance
• Measured in Henrys (H)
dIL
dtE
1 H 1 V s/A
L
Kirchoff’s rules for Inductors:• Assign currents to every path, as usual• Kirchoff’s first law is unchanged• The voltage change for an inductor is L (dI/dt)
• Negative if with the current• Positive if against the current
• In steady state (dI/dt = 0) an inductor is a wire
+–
LE
I
What is Kirchoff’s law for the loop shown?A) E + L (dI /dt) = 0 B) E – L (dI /dt) = 0 C) None of the aboveD) I don’t know Kirchoff’s law for switches
0 EdI
Ldt
dI
dt LE
I tL
E
![Page 4: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/4.jpg)
Ex- (Serway 32-9) The current in a 90.0 mH inductor changes with time as I = t2 - 6t (in SI units). Find the magnitude of the induced emf at (a) t = 1.00 s and (b) t = 4.00 s. (c) At what time is the emf zero.
Solve on Board
![Page 5: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/5.jpg)
Warmup 18
![Page 6: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/6.jpg)
Energy in Inductors• Is the battery doing work on the inductor?
I V P+–
LEdI
ILdt
• Integral of power is work done on the inductor
U dtPdI
IL dtdt
L IdI 212 LI k
• It makes sense to say there is no energy in inductor with no current21
2U LI• Energy density inside a solenoid?
2 20
2
N AIU
0NIB
20L N A
Uu
A
2 20
22
N I
2
02
B
2
02
Bu
• Just like with electric fields, we can associate the energy with the magnetic fields, not the current carrying wires
![Page 7: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/7.jpg)
RL Circuits• Circuits with resistors (R) and inductors
(L) L
E = 12 V
+–
In the steady state, with the switch closed, how much current flows through R2? How much current flows through R2 the moment after we open the switch?A) 0 A B) 6 A C) 3 AD) 2 A E) None of the above
R1 =
2
• In the steady state, the inductor is like a wire• Both ends of R2 are at the same potential: no current through R2
• The remaining structure had current I = E/R1 = 6 A running through it
6 A6 A
I = 6 A
R2 =
4
• Now open the switch – what happens?• Inductors resist changes in current, so the current
instantaneously is unchanged in inductor• It must pass through R2
I = E /R1 = 6 A
6 A
![Page 8: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/8.jpg)
RL Circuits (2)• What happens after you open the switch?
• Initial current I0
• Use Kirchoff’s Law on loop• Integrate both sides of the equation
L
E = 12 V
+–R1 =
2
I
R =
4
0dI
L RIdt
dI RI
dt L
dI Rdt
I L dI R
dtI L
ln constantRt
IL
Rt LI e
0tI I e
L R
![Page 9: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/9.jpg)
RL Circuits (3)• Where did the energy in the inductor go?• How much power was fed to the resistor?
L
E
+–
R
0tI I e
L R
• Integrate to get total energy dissipated
2 20 02
Rt LLRI e
R
20 2
LRI
R
2102U LI • It went to the resistor
• Powering up an inductor:• Similar calculation
1 tI eR
E
2RIP 2 20
tR LRI e
0
U dt
P 2 20
0
Rt LRI e dt
![Page 10: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/10.jpg)
Sample Problem
L =
4.0
mH
I R
0tI I e L R
An inductor with inductance 4.0 mH is discharging through a resistor of resistance R. If, in 1.2 ms, it
dissipates half its energy, what is R?
210 02U LI 21
2U LI 102 U 21
04 LI2 21
02fI I0
10.707
2
I
I
0
0.707tIe
I ln 0.707 0.347t
0.347
t 31.2 10 s
0.00346 s0.347
LR
.00400 H
.00346 sR 1.16 R
![Page 11: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/11.jpg)
Concept Question
L
E = 10 V
+–
The circuit at right is in a steady state. What will the voltmeter read as soon as the switch is opened?A) 0.l V B) 1 V C) 10 VD) 100 V E) 1000 V
R1 =
10
• The current remains constant at 1 A• It must pass through resistor R2
• The voltage is given by V = IR
R2 =
1 k
• Note that inductors can produce very high voltages
• Inductance causes sparks to jump when you turn a switch off
I =
1 A
1 A 1000 V IR
1000 VV
+–
Loop has unin-tended inductance
V
![Page 12: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/12.jpg)
CT-1- When the switch is closed, the current through the circuit exponentially approaches a value I=E/R. If we repeat this experiment with an inductor having twice the number of turns per unit length, the time it takes for the current to reach a value of I/2
A increases. B. decreases. C. is the same.
![Page 13: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/13.jpg)
CT-2 When the switch in the circuit below is closed, the brightness of the bulb
+ -
R
Bulb
L A. Starts off at its brightest and then dims. B. Slowly reaches its maximum brightness. C. Immediately reaches it maximum, constant brightness. D. Something else.
Assume inductor has no resistance
![Page 14: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/14.jpg)
Inductors in series and parallelL1• For inductors in series, the
inductors have the same current
• Their EMF’s add
L2
1
dIL
dtE 2
dIL
dt 1 2
dIL L
dt
1 2L L L
• For inductors in parallel, the inductors have the same EMF but different currents
L1
L2
11
dIL
dtE
22
dIL
dtE
1 2dI dIdI
dt dt dt
1 2L L
E EL
E
1 2
1 1 1
L L L
![Page 15: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/15.jpg)
Parallel and Series - Formulas
Capacitor Resistor Inductor
Series
Parallel
Fundamental Formula
1 2R R R
1 2
1 1 1
R R R 1 2C C C
1 2
1 1 1
C C C 1 2L L L
1 2
1 1 1
L L L
QV
C V IR
L
dIL
dtE
![Page 16: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/16.jpg)
Warmup 19
![Page 17: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/17.jpg)
CT - 3- The primary coil of a transformer is connected to a battery, a resistor, and a switch. The secondary coil is connected to an ammeter. When the switch is thrown closed, the ammeter shows
A. zero current. B. a nonzero current for a short instant. C. a steady current. D. Something else.
![Page 18: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/18.jpg)
LC Circuits• Inductor (L) and Capacitor (C)• Let the battery charge up the capacitorNow flip the switch• Current flows from capacitor through inductor• Kirchoff’s Loop law gives:• Extra equation for capacitors:
+–
EC
L
Q
0Q C V C EI
0Q
C dI
Ldt
dQI
dt
dIQ CL
dt d dQ
CLdt dt
2
2
1d QQ
dt CL
• What function, when you take two deriva-tives, gives the same things with a minus sign?
• This problem is identical to harmonicoscillator problem
cos
sin
Q t
Q t
0 cosQ Q t
![Page 19: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/19.jpg)
LC Circuits (2)• Substitute it in, see if it works
C
L
Q
I
0 cosQ Q t
0 sindQ
Q tdt
2
202
cosd Q
Q tdt
2
2
1d QQ
dt CL
20 0
1cos cosQ t Q t
CL
1
CL
• Let’s find the energy in the capacitor and the inductor
dQI
dt 0 sinQ t
2
2C
QU
C
2
20 cos2C
QU t
C
212LU LI 2 2 21
02 sinLQ t
2
20 sin2L
QU t
C
20 2C LU U Q C
Energy sloshes back and forth
![Page 20: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/20.jpg)
Warmup 19
![Page 21: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/21.jpg)
![Page 22: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/22.jpg)
Frequencies and Angular Frequencies• The quantity is called the angular frequency• The period is the time T you have to wait for it to repeat• The frequency f is how many times per second it repeats
2T
1
CL 0 cosQ Q t T
1f T
2 f
WFDD broadcasts at 88.5 FM, that is, at a frequency of 88.5 MHz. If they generate this with an inductor with L = 1.00 H,
what capacitance should they use?
2 f 6 12 88.5 10 s 8 15.56 10 s 2 1LC
2
1C
L
28 1 6
1
5.56 10 s 10 H
3.23 pFC
![Page 23: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/23.jpg)
Ex – Serway (32-49) A 1.00 µF capacitor is charged by a 40 V power supply. The fully-charged capacitor is then discharged through a 10.0 mH inductor. Find the maximum current in the resulting oscillations.
Solve on Board
![Page 24: Self Inductance Consider a solenoid L, connect it to a battery Area A, length l, N turns What happens as you close the switch? Lenz’s law – loop resists](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d8d5503460f94a75c06/html5/thumbnails/24.jpg)
RLC Circuits• Resistor (R), Inductor (L), and Capacitor (C)• Let the battery charge up the capacitorNow flip the switch• Current flows from capacitor through inductor• Kirchoff’s Loop law gives:• Extra equation for capacitors:
+–
EC
L
Q
I
0Q dI
L RIC dt
dQI
dt 2
20
Q dQ d QR L
C dt dt
• This equation is hard to solve, but not impossible• It is identical to damped, harmonic oscillator
20 cosRt LQ Q e t
R
2
2
1
4
R
LC L