announcements change of plans for today: demos on light and selected review for today
TRANSCRIPT
Announcements
Change of plans for today:Demos on light and selected review for today
Faraday’s Law
3 m/s 2 m
10 m
5 T10
What is the current induced in this circuit?
A) AB) AC) 10A
D) 6A
Faraday’s Law
Bd
dt
E
3 m/s 2 m
10 m
5 T10
As the bar moves a current is induced!
There are no batteries anywhere, so we say that a current is induced, by an induced emf.
Hence, an electric current can be induced in a circuit by a changing magnetic field, in the opposite direction to the change in flux.
Comparision of Induction
BdE ds
dt
0 0 0Ed
B ds Idt
•No magnetic monopole, hence no magnetic current•Electric fields and magnetic fields induce in opposite fashions
Faraday’s Law and Electric Fields
BdE ds
dt
. A cylindrical region of radius R = 3.0 cm contains a uniform magnetic field parallel to its axis. The field is 0 outside the cylinder. If the field is changing at the rate 0.60 T/s, the electric field induced at a point 2R from the cylinder axis is:
Using Faraday’s law: 2 (2R)E =-(R2) dB/dt, so E= (-(R2) /4) dB/dt=0.0045 V/m
Maxwell’s Equations
BdE ds
dt
Integral Form
0 0 0Ed
B ds Idt
0S
B dA
0
in
S
qE dA
Gauss’s laws, Ampere’s law and Faraday’s law all combined!
They are nearly symmetric with respect to magnetism and electricity.
The lack of magnetic monopoles is the main reasonwhy they are not completely symmetric.
Quiz
The diagrams show three circuits with identical batteries, identical inductors, and identical resistors. Just after the switch is closed which has the least current through the battery?
The diagrams show three circuits with identical batteries, identical inductors, and identical resistors. Just after the switch is closed which has the greatest current through the battery?
The diagrams show three circuits with identical batteries, identical inductors, and identical resistors. A very long time later, which has the least current through the battery?
RL - Circuits
E–
+
•What happens when the switch S is closed at t = 0? R
L
S
•Let I be the current in the circuit
I
•Use Kirchoffs rule for loops on the circuit
0 E RIdI
Ldt
dI R
Idt L L
E
) ) )/ /1 1Rt L tI t e eR R
E E
/L R
RC–Circuits vs RL-Circuits
L
tRe
RI 1
RCt
eR
I
•At t=0, ordinary wire
•As t-> infinity, broken wire•At t=0, broken wire, little currentfor small t
•As t-> infinity, ordinary wire
•In terms of current control, an inductor can often be considered as the opposite of a capacitor
)sinI C t E
LC – Circuits and Energy
+– L
C
S1S2
)cosV t E
1/ LC At an arbitrary time t, where is the energy stored in this circuit?A) In the capacitorB) In the inductorC) Alternately in the capacitor or the inductorD) What energy?
212LU LI
)212CU C V )2 21
2 cosC t E
)2 2 2 212 sinLC t E )2 21
2 sinC t E
212totU C E
LC - Circuits
+–
EL
C
S1S2
•Switch S1 is closed, then opened.•At t = 0, switch S2 is closed.•What happens?
)0V t E
Q C V dQ
Idt
I
dIV L
dt
2
2
d QL
dt )2
2
d VCL
dt
LC – Circuits and Harmonic Oscillators
)2
2
d VV CL
dt
)cosV t E
These equations
2
2
dt
xd
k
mx
)cos( tAx
There are many correspondances between electrical and mechanical systems!
RLC circuits in Series IIL CR
S
RIdt
dU 2
22
22 LI
C
QU
dt
dILI
dt
dQ
C
Q
dt
dU
Do some algebra, and use
dt
dQI
02
2
C
Q
dt
dQR
dt
QdL
RLC circuits and Harmonic Oscillators
LC
R
S
RIdt
dU 2
A damped harmonic oscillator!
02
2
C
Q
dt
dQR
dt
QdL
02
2
kxdt
dxb
dt
xdm
Hence, the charge oscillations are the same as the motion of a damped harmonic oscillator.
Quiz
A.B.
C.
D.
Electromagnetic Waves
0 0kE Bck 0 0E cB y zE cB
)0 cosyE E kx t )0 coszB B kx t
Electric FieldMagnetic
Field
Direction of Motion
y x zE E B
x y t
yx zBE E
z x t
y xzE BE
y z t
0 0y x z
B B E
x y t
0 0yx z
EB B
z x t
0 0y xz
B EB
y z t
Using Maxwell’s Equations
y zE B
x t
0 0
yzEB
x t
Electromagnetic Waves
•These equations look like sin functions will solve them.
)0 cosyE E kx t )0 coszB B kx t
) ) ) )
0 0
0 0 0 0
sin sin
sin sin
kE kx t B kx t
kB kx t E kx t
0 0 0 0 0 0 kE B kB E
Electromagnetic Waves
0 0 0 0 0 0 kE B kB E
•These equations imply
2 20 0 0 0 0 0k E B B E
2
20 0
1
k
0 0
1
k
82.998 10 m/sc
•The speed of light (in vacuum)