Download Lecture 28 Chapter 33 EM Oscillations and AC

Lecture 28
Chapter 33
EM Oscillations and AC
Review
r r
dΦE
∫ B • ds = µ0ε0 dt + µ0ienc
• Can represent change in electric flux
with a fictitious current called the
displacement current, id
dΦ
id = ε 0
• Ampere-Maxwell’s law becomes
r r
∫ B • ds = µ0id ,enc + µ0ienc
dt
E
Review
Maxwell’s 4 equations are
• Gauss’ Law
∫
r q enc
r
E • dA =
ε0
• Gauss’ Law for magnetism
r r
∫ B • dA = 0
r r
dΦ B
• Faraday’s Law ∫ E • ds = − dt
r r
d
Φ
E
• Ampere-Maxwell Law B • ds = µ0ε 0
+ µ0ienc
∫
dt
EM Oscillations (1)
• In this chapter study an RLC circuit
• Review RC and RL circuits
• Charge, current and potential difference
grow and decay exponentially given by
time constant, τC or τL
• Look at LC circuit
EM Oscillations (2)
• RC circuit - resistor &
capacitor in series
– Charging up a capacitor
q = CE (1 − e
−t τ c
– Discharging capacitor
q = q0 e
– where
−t τc
τ C = RC
)
EM Oscillations (3)
• RL Circuit – resistor &
inductor in series
– Rise of current
E
−t τ L
i = (1 − e )
R
– Decay of current
i = i0e
– where
τL
−t τ L
L
=
R
EM Oscillations (4)
• LC Circuit – inductor &
capacitor in series
• Find q, i and V vary
sinusoidally with period T
and angular frequency ω
• E field of capacitor and B
field of inductor oscillate
• Called Electromagnetic oscillations
• Energy stored in E of capacitor and B of
inductor
2
2
q
UE =
2C
U
B
Li
=
2
EM Oscillations (5)
EM Oscillations (6)
• Cycle repeats at
some frequency, f
and thus angular
frequency, ω
ω = 2π f
• Ideal LC circuit, no
R so oscillations
continue indefinitely
• Real LC, oscillations
die away as energy
goes into heat in R
EM Oscillations (7)
• Checkpoint #1 – A charged capacitor & inductor
are connected in series at time t=0. In terms of
period, T, how much later will the following reach
their maximums:
– q of capacitor
T/2
– VC with original polarity
T
– Energy stored in E field
T/2
– The current
T/4
EM Oscillations (8)
• LC circuits analogous
to block-spring system
• Total energy of block
U = mv + kx
1
2
2
1
2
2
• Energy is conserved dU
• Differentiating gives
dt
=0
dx
dU d 1 2 1 2
dv
= (2 mv + 2 kx ) = mv
+ kx
=0
dt
dt
dt
dt
EM Oscillations (9)
dx
v =
dt
• Using
• Substitute
• Gives
dv
d 2x
=
dt
dt 2
2
d x
dx
dv
+ kx
= mv 2 + kxv = 0
mv
dt
dt
dt
2
d x
m
+ kx = 0
2
dt
• Solution is
x = X cos( ωt + φ )
• X is the amplitude
• ω is the angular
frequency
k
• φ is the ω =
m
phase
constant
EM Oscillations (10)
• Total energy of LC circuit
Li 2 q 2
+
U = UB +UE =
2
2C
• Total energy is constant
dU
=0
dt
• Differentiating gives
dU d  Li
q 
di q dq
 = Li +
= 
+
=0
dt
dt  2
2C 
dt C dt
2
2
EM Oscillations (11)
• Using
dq
i=
dt
• Substitute
di d 2 q
= 2
dt dt
2
d q q
di q dq
Li +
= Li 2 + i = 0
dt C dt
dt
C
• Equation same form as
block and spring 2
• Solution is
d q 1
L 2 + q=0
dt
C
q = Q cos(ωt + φ )
• Q is the
amplitude
• ω is the angular
frequency
• φ is the phase
constant
EM Oscillations (12)
• Charge of LC circuit
q = Q cos(ωt + φ )
• Find current by
dq
i=
dt
d
i = [Q cos(ωt + φ )] = −Qω sin(ωt + φ )
dt
• Amplitude I is
I = ωQ
i = − I sin(ωt + φ )
EM Oscillations (13)
• What is ω for an LC circuit?
q = Q cos(ωt + φ )
• Substitute into
d 2q
2
=
−
ω
Q cos( ω t + φ )
2
dt
2
d q 1
L 2 + q=0
dt
C
1
− Lω Q cos( ωt + φ ) + Q cos( ω t + φ ) = 0
C
2
• Find ω for LC circuit is
ω=
1
LC
EM Oscillations (14)
• The phase constant, φ, is determined
by conditions at any certain time, t
q = Q cos(ωt + φ ) i = − I sin(ωt + φ )
• If φ = 0 at t = 0 then
q=Q
i=0
EM Oscillations (15)
• The energy stored in an LC
circuit at any time, t
• Substitute
U
U
• Using
B
E
U = UB + UE
q = Q cos(ωt + φ ) i = − I sin(ωt + φ )
q2
Q2
=
=
cos
2C
2C
2
(ω t + φ )
Li 2
L 2 2
=
=
ω Q sin
2
2
ω=
1
LC
U
B
2
(ω t + φ )
Q2
=
sin
2C
2
(ω t + φ )
EM Oscillations (16)
• For the case where φ = 0
U
U
E
B
Q2
=
cos
2C
Q2
=
sin
2C
2
(ω t + φ )
2
(ω t
+φ
• Maximum value for both
)
U E ,max = U B ,max = Q 2C
• At any instant, sum is U = U B + U E = Q2 2C
• When UE = max, UB = 0, and conversely,
when UB = max, UE = 0
2
EM Oscillations (17)
• Checkpoint #2 – Capacitor in LC circuit has
VC,max = 17V and UE,max = 160µJ. When
capacitor has VC = 5V and UE = 10µJ , what
are the a) emf across the inductor and b) the
energy stored in the B field, UB ?
• Can apply the loop rule
– Net potential difference around the
circuit must be zero
vL (t ) = vc (t )
A) vL = 5V
UE,max = UE (t) +UB (t)
B) UB = 160-10=150µJ
EM Oscillations (18)
• Consider a RLC circuit – resistor,
inductor and capacitor in series
• Total electromagnetic energy,
U = UE + UB, is no longer
constant
• Energy decreases with time as it
is transferred to thermal energy in
the resistor
• Oscillations in q, i and V are
damped
– Same as damped block and spring
EM Oscillations (19)
• Resistor does not store
electromagnetic energy so
total energy at any time is
2
2
Li
q
+
U = UB +UE =
2
2C
• Rate of transfer to thermal
dU
2
=
−
i
R
energy is (minus sign means
dt
U is decreasing)
• Differentiating gives
di q dq
dU
2
= Li +
= −i R
dt
dt C dt
EM Oscillations (20)
di q dq
dU
2
= Li +
= −i R
dt C dt
dt
dq
• Use relations i =
dt
2
di d q
= 2
dt dt
• Differential equation for damped RLC circuit is
2
dq
1
d q
L
+ R
+
q = 0
2
dt
dt
C
• Solution
q = Qe
− Rt / 2 L
cos(ω ′t + φ )
EM Oscillations (21)
− Rt / 2 L
q = Qe
cos(ω ′t + φ )
• Where
ω′ = ω − ( R / 2L)
2
2
ω =
1
LC
• Charge in RLC circuit is sinusoidal but with an
exponentially decaying amplitude
− Rt / 2 L
Qe
• Damped angular frequency, ω´, is always less
than ω of the undamped oscillations
EM Oscillations (22)
q = Qe
− Rt / 2 L
cos(ω ′t + φ )
• Find UE as function of time
2
2
q
Q − Rt / L
UE =
=
e
cos2 (ω ′t + φ )
2C 2C
2
• Total energy decreases as
U tot
Q − Rt / L
=
e
2C