Transcript Slide 1

-Mutual Inductance
-LC Circuit
AP Physics C
Mrs. Coyle
• Induced emf:
dI
εL   L
dt
Inductance
εL
L
d I dt
Mutual Inductance
• Two circuits interact
and the varying
magnetic flux in one
circuit causes an
induced emf in the
other circuit.
N212
M12 
I1
d 12
d I1
ε 2   N2
 M12
dt
dt
Mutual Inductance
• In mutual induction, the emf induced in one coil is
always proportional to the rate at which the current
in the other coil is changing
• The mutual inductance in one coil is equal to the
mutual inductance in the other coil
M12 = M21 = M
• The induced emf’s can be expressed as
d I2
ε1  M
dt
d I1
and ε2  M
dt
• Assume the capacitor is
initially charged and then
the switch is closed
LC Circuits
• Assume no resistance
(no internal energy loss)
and no energy losses to
radiation
• The total energy of this
system is constant
• The current in the
system oscillates similar
to a mass-spring system
Q2 1 2
U  UC  UL 
 LI
2C 2
Energy in an LC Circuit – Graphs
Charge and Current in an LC Circuit
• The current in the inductor
oscillates between Imax and -Imax
• Q and I are 90o out of phase
with each other
– So when Q is a maximum, I is zero,
etc.
LC Circuit Analogy to Spring-Mass
Q2 1 2
U  UC  UL 
 LI
2C 2
1 2 1 2
U  mv  kx
2
2
At t = ¼ T, all the energy is stored as magnetic energy in the inductor.
The maximum current occurs in the circuit
This is analogous to the mass at equilibrium
Charge and Angular Frequency of an
LC Circuit
Q = Qmax cos (ωt + φ)
• The angular frequency, ω, of the circuit depends on
the inductance and the capacitance
– It is the natural frequency of oscillation of the circuit
ω 1
LC
Time Functions of an LC Circuit
• Current as a function of time
dQ
I
 ωQmax sin( ωt  φ )
dt
• Total energy as a function of time:
2
Qmax
1 2
2
U  UC  UL 
cos ωt  L I max sin 2 ωt
2c
2