# ***RC circuits�

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```					***RC circuits…
What is the connection
between a pacemaker and
windshield wipers?
***Charging a capacitor
***Capacitors: Air tank analogy to
charging a capacitor
How many time constants does it take to fully charge
a capacitor?
Discharging a capacitor:
What is the current right
after the switch is closed?
Assuming the capacitor is initially discharged, what
is the current right after the switch is closed?
After a long time?
What is the voltage across this resistor
right after switch AB is closed?
What is the current right after the switch is
closed? After a long time?
RC circuit: Q and I vs time
***RC circuit: Q and I vs time
Determine the current as a function of time after
the switch is closed.

I = 200µA + e(-t/1s)(125µA)
Resistive-capacitive time constant
Applet: RC time constant
Applet: RC circuit
If the capacitor is initially uncharged, what is the
charge after a long time? How long will it take the
capacitor to reach 80% of full charge?

Q  CV  (182 F)(3V)  546 C
Q(t)/Q  1 - e-t/RC  0.8
t  118 ms
***This is a simplified circuit for a photographic flash unit. This circuit
consists of a 9-V battery, an R = 50-kΩ resistor, a 140 µF capacitor, a
flashbulb, and two switches. Initially, the capacitor is uncharged and the
two switches are open. To charge the unit, switch S1 is closed; to fire the
flash, switch S2 (which is connected to the camera's shutter) is closed.
How long does it take to charge the capacitor to 4.6 V? How much energy
is released when the lamp flashes? What fraction of the energy supplied
by the battery is dissipated in the resistor during the charging process?

V(t)/Vm ax  1 - e -t/RC  4.6/9
t/RC  0.716
t  5.0 s

Q  CVc  (140 F)(4.6V)  644 C
E c  1 CVc2  1  Q  4.6  1.48 mJ
2        2

E b  Q  Vb  644 C  9  5.8 mJ
Er  Eb  Ec 
Suppose there is a current through the lamp only when the potential
difference across it reaches the breakdown voltage VL; in this event,
the capacitor discharges completely through the lamp and the lamp
flashes briefly. Suppose that two flashes per second are needed. For a
lamp with breakdown voltage VL = 63 V, wired to a 98 V ideal battery
and a 0.15 μF capacitor, what should be the resistance R?

Vc = ε∙(1 – exp(-t/RC))
1 – Vc/ε = exp(-t/RC)
t/RC = ln(ε/(ε – Vc))

t
R                      
C·ln(/( – Vc ))
0.5
                                3.24x106 
0.1510-6 ·ln(98/(98 – 63))
In this simple "sawtooth oscillator”, the neon bulb initially has infinite
resistance until the voltage across it reaches 90 V, and then it begins to
conduct with very little resistance (essentially zero). It stops
conducting when the voltage drops down to 70 V. If ε = 100 V, R = 32
kΩ, C = 3.85 µF, what is the frequency of the flashing?

70/100  1 - e - t1    RC
; t1  148 ms

90/100  1 - e - t 2   RC
; t 2  284 ms
t = t2-t1 = 137 ms
f = 1/t = 7.3 Hz
***The figure below shows the circuit for a simple "sawtooth oscillator",
where R = 32.0 kΩ and C = 4.10 μF. At time t = 0, its switch S is closed.
The neon bulb has initially infinite resistance until the voltage across it
reaches 90 V, and then it begins to conduct with very little resistance
(essentially zero). It stops conducting (its resistance becomes essentially
infinite) when the voltage drops down to 66 V.
Electronic devices often use an RC circuit to protect against power
outages, as shown. If the device is supposed to keep the supply voltage at
least 75% of nominal for as long as 0.2 s with a 12 µF, how big a resistance
is needed?

V f / V0  e  t / RC  0.75
RC  0.695 s
R  0.695 / 12  10 6  57 .9 k
*What is the potential at points a and b with S open? What is
the final potential at these points after S is closed? How much
charge flows through S after closure?

c

open:
C1        Va = 4.4/(4.4+8.8) = 8 V
Q1 = Q2 = V∙Ceq = 3.84 μC
Vb = Q2/C2 = 16 V
closed:
Va = Vb = 8 V
Q1 = C1∙Vcb = 0.48∙16 = 7.68 μC
C2        Q2 = C2∙Vbd = 0.24∙8 = 1.92 μC
Qb = -Q1 + Q2 = -7.68 + 1.92 = -5.8 μC

d
*What is the time constant (or constants) for
charging the capacitors after the 24 V is applied?

Vcd  [(R1  R2 ) / R1C1 )]Q1  [ R2 (C1  C2 ) / C1 ]dQ1 / dt
compare :   Q / C  RdQ / dt
  R1R2 (C1  C2 ) /( R1  R2 )  2.1 s
Electrocardiographs are often connected as shown. The leads are said to
be capacitively coupled. A time constant of 3.0 s is typical and allows rapid
changes in potential to be accurately recorded. If C = 2.4 µF, what value
must R have?
***Capacitor: The Earth's upper atmosphere interacts with
high energy particles from the sun (called the solar wind) to
produce a steady stream of positive ions, which, in turn, rain
down onto the Earth's surface. Yet, the Earth's total average
surface charge does not change. Why?
We can model the Earth’s atmosphere as the positive plate of a giant
capacitor, with the negative plate being the surface of the Earth, on
which about 5x105 C is spread. If the separation of the plates is
taken to be about 5 km, the resistivity of air is 3x1013 Ω·m, and a
typical lightning strike delivers about 25 C of negative charge to the
ground (this is the charging process), approximately how many
lightning strokes are there per day?

R = ρ·l/A = 3x1013·5000/(4π·6.6x106)2 = 300 Ω
C = 0.9 F
= RC = 300 s = 5 min
< 0.3% would remain after 30 minutes
Atmospheric charge = 5x105 C
Charge per stroke = 25 C
#strokes for recharge = 5x105/25 = 20,000
Time for recharge ~ 30 minutes
#strokes per day = 20,000/30 min = 1 million
End
***LRC circuits…
How do spark plugs generate such huge voltages?
Applet: RL current build-up
What is the time dependence of the current in this
LR circuit?

dI
 L        IR  0
dt
I  I   (1  e t / L )
I    / R;  L  L / R
Current vs time in an RL circuit

dI
dI                                L         IR  0
  L  IR  0                                dt
dt
I  I 0  e t /  L
I  I   (1  e t / L )
L  L/ R
I    / R;  L  L / R
What does the current look like if the resistance is
very small?
What is the voltage across the light bulb a long time
after the switch is closed? What if you now open the
switch?
When the switch is closed, the current in the circuit is
observed to increase from 0 to 0.2 A in 0.15 seconds. What is
the maximum current in the circuit? How long after the switch
is closed does the current have 0.4 A?

I max  I   /R  9/5.5  1.64 A

  L·dI/dt
L  /(dI/dt)  9/(0.2/0.1  6.75 H
5)
 L  L/R  1.23 s

I  I  ·(1 - e -t/1.23 )  0.4 A
t  0.34 s

I  I ·(1- e-t/ L )  0.2 A
 L  1.15 s

I  I  ·(1 - e -t/1.15 )  0.4 A
t  0.32 s
If the emf=12 V, R=150 Ω, and L=56 mH, what is the
current right after and a long time after the switch
is closed?
The circuit shown consists of a 4.5 V battery, a 37-mH
inductor, and four 55-Ω resistors. What is the current through
the inductor immediately after and a long time after the switch
is closed? What is the current through the inductor two
characteristic time intervals after closing the switch.

τ = L/Req = L/(5R/3) = 0.404 ms
I0 = 0 A; I∞ = V/(5R/3) = 49 mA
I(2τ) = I∞∙[1-exp(-t/τ)] = 42.4 mA
After the switch is closed for a long time, the
energy stored in the inductor is 3 mJ. What is the
value of R?
A very large, superconducting solenoid such as the one used in MRI
scans has a self-inductance of 150 H and can carry currents as large
as 100 A. If the coils suddenly go “normal”, what temperature increase
is produced if all the stored energy goes into heating the 1000-kg
magnet, given its average specific heat is 200 J/kg·ºC

E  1 LI 2  750 ,000 J
2

750 ,000  200 1000  T
T  3.75 C 

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