Battery

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Capacitors
Short-term charge stores
Battery
Capacitor
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• Capacitance is defined as
• The charge required to cause unit potential
  difference in a conductor.
• Capacitance is measured in units called farads
  (F) of which the definition is
• 1 Farad is the capacitance of a conductor, which
  has potential difference of 1 volt when it
  carries a charge of 1 coulomb.

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• So we can write
• Capacitance (F) Charge (C)
• Voltage (V)
• In code, this is written
• C Q
• V
• Q - charge in coulombs (C) C capacitance in
  farads (F) V - potential difference in volts (V)

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• A 1 farad capacitor is actually a very big
  capacitor indeed so instead we use microfarads
  (mF) where 1 mF 1 10-6 F.
• Smaller capacitors are measured in nanofarads
  (nF), 10-9 F, or picofarads (pF), 1 10-12 F.
• A working voltage is also given. If the
  capacitor exceeds this voltage, the insulating
  layer will break down and the component shorts
  out. The working voltage can be as low as 16
  volts, or as high as 1000 V.

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• If we connect a capacitor in series with a bulb
• If connected to a d.c. circuit, the bulb flashes,
  then goes out.
• In an a.c. circuit, the bulb remains on.
• We can say that a capacitor blocks d.c., but
  allows a.c. to flow.

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• When we charge up a capacitor, we make a certain
  amount of charge move through a certain voltage.
  We are doing a job of work on the charge to build
  up the electric field in the capacitor. Thus we
  can get the capacitor to do a job of useful work.
  
• We know that
• Energy charge voltage
• Q CV.
• This second relationship tells us that the charge
  voltage graph is a straight line.

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Energy in a Capacitor

• The capacitor is charged with charge Q to a
  voltage V. Suppose we discharged the capacitor
  by a tiny amount of charge, dQ. The resulting
  tiny energy loss (dW) can be worked out from the
  first equation
• dW V dQ
• This is the same as the area of the grey
  rectangle on the graph.

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• If we discharge the capacitor completely, we can
  see that
• Energy loss area of all the little rectangles
• area of triangle below the graph
• ½ QV
• By substitution of Q CV, we can go on to
  write

E ½ CV2
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Exponential Discharge of a Capacitor

• If it takes time t for the charge to decay to 50
  of its original level, we find that the charge
  after another t seconds is 25 of the original
  (50 of 50 ).
• This time interval is called the half-life of
  the decay. The decay curve against time is
  called an exponential decay.
•  The voltage, current, and charge all decay
  exponentially during the capacitor discharge.

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Circuit to measure discharge of a capacitor
Note that we can leave out the voltmeter or the
ammeter
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Exponential Decay of a Capacitor
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• We should note the following about the graph
•         Its shape is unaffected by the voltage.
•         The half life of the decay is
  independent of the voltage.
•         The current follows exactly the same
  pattern as I V/R.
•         The charge is represented by the
  voltage, as Q CV.
•         The graph is asymptotic, i.e. in theory
  the capacitor does not completely
  discharge. In practice, it does.

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The graph is described by the relationship   Q
Q0 e t/RC   Q charge (C) Q0 charge at the
start e exponential number (2.718) t time
(s) C capacitance (F) R resistance
(W).   For voltage and current, the equation
becomes         V V0 e t/RC         I I0 e
t/RC
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• The product RC (capacitance resistance) which
  we see in the formula is called the time
  constant.
• The units for the time constant are seconds. We
  can go back to base units to show that ohms
  farads are seconds.
• So if we discharge the capacitor for RC
  seconds, we can easily find out the fraction of
  charge left
• V V0 e RC/RC V0 e 1 0.37 V0 
• So after RC seconds the voltage is 37 of the
  original. This is used widely by electronic
  engineers. To increase the time taken for a
  discharge we can
•         Increase the resistance.
•         Increase the capacitance.

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We can link the half-life to the capacitance. At
the half life         Q Q0/2         t
t1/2 Q0/2 Q0 e t1/2/RC Þ ½ e t1/2/RC Þ
2-1 e t1/2/RC Þ e t1/2/RC 2 Þ loge (2)
t1/2/RC In text books you may see the
natural logarithm written as ln Þ t1/2
loge (2) RC 0.693 RC  The half-life is 69
of the time constant.