Download Physics for Scientists & Electromagnetic Oscillations !

Electromagnetic Oscillations
! We have been working with circuits that have
• a constant current
• a current that increases to a constant current
Physics for Scientists &
Engineers 2
• a current that decreases to zero current
! Now we will introduce circuits that have resistors,
capacitors, and inductors as well as time-varying sources of
emf
Spring Semester 2005
! These types of circuits display different phenomena than
the circuits we have been studying, e.g.
Lecture 28
• Resonance
• Transformers
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Physics for Scientists&Engineers 2
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Energy in Inductors
! Remember that the energy stored in the electric field of a
capacitor with charge q is given by
• The current increases or decreases exponentially with a time
constant !C = RC
UE =
! Consider a circuit with an inductor L and a capacitor C
• The current increases or decreases exponentially with a time
constant !L = L/R
1 q2
2C
! Remember that the energy stored in the magnetic field of
an inductor carrying current i is given by
! Now consider a circuit with an inductor L and a capacitor C
• We will see that the current and voltage will vary sinusoidally with
time
UB =
! We call these variations in voltage and current
electromagnetic oscillations
Physics for Scientists&Engineers 2
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Energy in Inductors (2)
! Consider a circuit with a resistor R and a capacitor C
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Physics for Scientists&Engineers 2
1 2
Li
2
! To illustrate electromagnetic oscillations, consider a simple
circuit containing a capacitor C and an inductor L
3
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LC Circuit Time
Evolution
LC Circuit
! The capacitor is initially fully charged and then connected to the circuit
! The energy in the circuit resides solely in the electric field of the
capacitor
! The current is zero
! Now let’s follow the evolution with time of the current, charge,
magnetic energy, and electric energy in the circuit
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LC Circuits (2)
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LC Circuits (3)
! The charge on the capacitor varies with time
• Max positive to zero to max negative to zero back to
max positive
! The current in the inductor varies with time
• Zero to max positive to zero to max negative back to
zero
! The energy in the inductor varies with the square
of the current and the energy in the capacitor
varies with the square of the charge
• The energies vary between zero and a maximum value
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LC Oscillations
LC Oscillations (2)
! Now we derive a quantitative description of the phenomena
we just described qualitatively
! We assume a single loop circuit containing a capacitor C and
an inductor L and that there is no resistance in the circuit
! We can write the energy in the circuit U as the sum of the
electric energy in the capacitor and the magnetic energy in
the inductor
dU d ! 1 q 2 1 2 $ q dq
di
= #
+ Li & =
+ Li = 0
dt
dt " 2 C 2
dt
% C dt
! We can re-write the electric and magnetic energies in
terms of the charge q and current i
1 q2 1 2
U = UE + UB =
+ Li
2C 2
Physics for Scientists&Engineers 2
! Thus the derivative of the energy in the circuit with
respect to time will be zero
! We can then write
U = UE + UB
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! Because we have assumed that there is no resistance and
the electric field and magnetic field are conservative, the
energy in the circuit will be constant
! Remembering that i = dq/dt we can write
di d ! dq $ d 2 q
= # &=
dt dt " dt % dt 2
9
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Physics for Scientists&Engineers 2
LC Oscillations (3)
LC Oscillations (4)
! We can then write
q dq
dq d 2 q q
d 2q
+L
=
+
L
=0
C dt
dt dt 2 C
dt 2
! The solution to the differential equation describing
simple harmonic motion was
! Which we can rewrite as
! where " is the phase and #0 is the angular frequency
x = xmax cos (! 0t + " )
d 2q
q
+
=0
2
dt
LC
!0 =
Physics for Scientists&Engineers 2
k
m
! By analogy we can get the charge as a function of time
! This differential equation has the same form as that of
simple harmonic motion describing the position x of an
object with mass m connected to a spring with spring
constant k
d2x k
+ x=0
dt 2 m
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q = qmax cos (! 0t + " )
! where " is the phase and #0 is the angular frequency
!0 =
11
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1
=
LC
1
LC
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LC Oscillations (5)
LC Oscillations (6)
! The current is then given by
! Having expression for the current as a function of time we
can write and expression for the magnetic energy
dq d
i=
= ( qmax cos (! 0t + " )) = #! 0 qmax sin (! 0t + " )
dt dt
UB =
! Realizing that the magnitude of the maximum current in
the circuit is given by #0qmax we get
! Remembering that
i = !imax sin(" 0t + # )
imax = ! 0 qmax
! Having expression for the charge as a function of time we
can write and expression for the electric energy
UE =
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2
max
Physics for Scientists&Engineers 2
1
LC
L 2
L
q2
2
imax = ! 02 qmax
= max
2
2
2C
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LC Oscillations (7)
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LC Circuit Example
! Now we can write an expression for the magnetic energy as a function
of time
UB =
!0 =
! We can write
1q
1 ( qmax cos (! 0t + " ))
q
=
=
cos 2 (! 0t + " )
2C 2
C
2C
2
2
1 2 L
L 2
2
Li = ( !imax sin(" 0t + # )) = imax
sin 2 (" 0t + # )
2
2
2
2
max
q
sin 2 (! 0t + " )
2C
! We can write an expression for the energy in the4 circuit by adding the
electric energy and the magnetic energy
2
qmax
q2
cos 2 (! 0t + " ) + max sin 2 (! 0t + " )
2C
2C
2
q
q2
U = max sin 2 (! 0t + " ) + cos 2 (! 0t + " ) = max
2C
2C
! Consider a circuit containing a capacitor
C = 1.50 µF and an inductor L = 3.50 mH.
The capacitor is fully charged using a
battery with Vemf = 12.0 V and then
connected to the circuit.
U = UE + UB =
(
! What is the oscillation frequency of the circuit?
)
! What is the energy stored in the circuit?
! What is the charge on the capacitor after t = 2.50 s?
! Thus the energy in the circuit remains constant with time and is
proportional to the square of the original charge put on the capacitor
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LC Circuit Example (2)
LC Circuit Example (3)
! The oscillation frequency of the circuit
is given by
1
=
LC
!0 =
( 3.50 "10
1
#3
)(
H 1.50 "10
#6
F
! The maximum charge on the capacitor
is given by
(
)
qmax = 1.80 !10 "5 C
! 0 = 1.38 "10 Hz
5
(
2
2
qmax
2C
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LC Circuit Example (4)
! The charge on the capacitor as a
function of time is given by
q = qmax cos (! 0t + " )
at t = 0, q = qmax # " = 0
q = qmax cos (! 0t )
qmax = 1.80 $10 %5 C
! 0 = 1.38 $10 5 Hz
at t = 2.50 s, we have
) (
(
)
q = 1.80 $10 %5 C cos &'1.38 $10 5 Hz () [ 2.50 s ]
q = -1.21$10 C
-5
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)
1.80 !10 "5 C
q2
U = max =
= 1.08 !10 "4 J
2C 2 !1.50 !10 "6 F
! The energy stored in the circuit is given by
U=
)
qmax = CVemf = 1.50 !10 "6 F (12.0 V)
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