AC circuits
Physics 114
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Lecture IX
1
Math review
• Integrals and derivatives of trig. functions:
d sin( t )
  cos(t )
dt
d cos(t )
  sin( t )
dt
 sin( t )dt  
 cos(t )dt 
1

1

cos(t )
sin( t )
• Relations between trig functions:
sin( t  90 )  cos(t )
cos(t  90 )   sin( t )
sin( t  90 )   cos(t )
cos(t  90o )  sin( t )
o
o
o
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Lecture IX
2
Self inductance
• Magnetic field in a solenoid
N
B  0 I
l
• It creates a magnetic flux
through itself
N
  BA   0 IA
l
• Self inductance of a solenoid
2
N 0 N A
L

I
l
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Lecture IX
In general by definition:
B
LI  N B
LN
I
d B
dI
Ε -N
 -L
dt
dt
3
Direction of induced emf
• In accordance to Lenz law
• I – increase  increase  induced magnetic field in
the opposite direction to initial magnetic field  emf in
the opposite direction to original emf
• I – decrease  decrease  induced magnetic field in
the same direction to initial magnetic field  emf in the
same direction to original emf
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Lecture IX
4
Current and voltage in AC circuit
VR  RI
Drop of voltage over
resistor (V) follows I
Current and voltage in phase
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Lecture IX
5
Current and voltage in AC circuit
Drop of voltage = - emf
dI
VL  L
dt
1. I goes up – V>0 – loose voltage
2. I goes down – V<0 – gain voltage
Inductor: current lags voltage
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Lecture IX
6
Current and voltage in AC circuit
Q 1
VC    Idt
C C
1. t=0, current flows to capacitor
 gain charge  gain voltage
2. Current changes sign
 drain charge  loose voltage
Capacitor: voltage lags current
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Lecture IX
7
Energy in AC circuit
2
1 2 1  0 N A  Bl 
1 B2
 

U L  LI  
Al
2
2  l  0 N 
2 0
1 B2
u  energy density 
2 0
2
dI
P  IV  LI
dt
dW  Pdt  LIdI
Energy of the magnetic field stored in
an inductor
1 2
U L  LI
2
I
1 2
W   dW   LIdI  LI
2
0
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Lecture IX
8
Energy in AC circuit
CV
UC 
2
2
2
CV
A
1
2
2
Ed    0 E Ad
UC 
 0
2
2d
2
1
u  energy density   0 E 2
2
Energy of the electric field can be stored in a capacitor
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Lecture IX
9
LC circuit
• Two forms of energy:
– Electric – in a capacitor
– Magnetic - in solenoid
• Analogy with a mass on a
spring, two forms of
energy
– Kinetic
– Potential
• Oscillator!
• No energy is lost, it is just
changing its form
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Lecture IX
10
LC circuit
• Harmonic oscillations in the AC circuit.
• No battery. Start with a charge on the capacitor.
Q  Q0Cos(t )
Q Q0
VC  
Cos(t )
C C
• Once the capacitor starts discharging, the current will
flow through the inductor:
dQ
I
dt
 Q0Sin (t )
dI
VL  L   LQ0 2Cos(t )
dt
• The net drop of voltage over closed circuit must be
zero: VL  VC  0
Q0
 LQ0 Cos(t )  Cos(t )  0
C
2
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Lecture IX
1
 
LC
1 1
f 
2 LC
2
11
Energy in AC circuit
Power dissipated:
P=I2R=RI20cos2wt
Energy dissipated
PI R
2
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Lecture IX
12
LCR circuit
• Resistors dissipate energy
– Convert electrical energy
into thermal energy
• Resistor acts like friction
for a weight on a spring
• Damped oscillator!
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Lecture IX
13