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Capacitance
Year 13
What is a capacitor?
• A device which stores energy in an electric
field
• Two conductors insulated from one
another
Capacitors
Charging a capacitor
• When connected to a voltage source,
electrons flow onto one plate and off the
other, producing a charge difference
• The higher the charge already on the
plate, the harder it is to add even more
Charging a capacitor
• Constant voltage power supply
• As the capacitor becomes charged, the
current drops
V
V
A
B
+q
-q
Electron flow
+Q
-Q
Fully charged. No current
Capacitance
• We define the capacitance of a capacitor
as the charge stored per unit potential
difference:
Q
C  , or Q  CV
V
• The unit of capacitance is the Farad (F)
• Most devices have values ranging from pF
to mF
• Real devices also have a defined
maximum voltage, above which they break
down
Air tank analogy
Air tank analogy
Discharging a capacitor
• When a charged capacitor is connected to a
resistance, electrons flow to rebalance the
charge distribution
• As the charge difference reduces, so the
potential difference drops
V
A
Capacitance of a parallel plate
capacitor
[not examinable]
– e0 is the permitivity of free space
– K is the relative permitivity of any dielectric
between the plates
Capacitance modified by dielectric
[not examinable]
• Applied field polarises
the dielectric.
• This decreases the
effective field between
the plates.
• Easier to add more
charge - this increases
the capacitance.
• Note that the dielectric
should be an insulator!
Capacitors store energy
• How do we know that the
charged capacitor stores
energy?
• How did the capacitor gain
its energy store?
V
Energy stored by a capacitor
• It gets progressively harder
to add more charge to an Potential
difference
already charged capacitor
V
• Energy stored = work done
charging
• Work done adding DQ is
VDQ
• Work done charging
capacitor to V is area under
graph
1
1
E  QV  CV 2
2
2
DQ
Charge Q
Energy supplied by battery
• V=E/Q, so Energy from battery=QV
• Energy stored by capacitor =1/2 QV
• So where did the other 1/2 go?
• Dissipated by parasitic resistance
– This is always true, regardless of the value fo
the resistance
Example
• A 10 mF capacitor is charged to 20 V.
How much energy is stored?
• Energy stored = 1/2 CV2 = 2000 mJ
• Calculate the energy stored at 10 V (i.e. at
half the voltage):
• 500 mJ, one quarter of the previous value
(EaV2)
Charging and discharging
capacitors
Capacitor Discharge
• Switch to A: capacitor charges
• Switch to B: capacitor discharges
• At a time t, charge is Q, pd is V and
current is I:
I=V/R around circuit
I=-DQ/Dt in a short time Dt
V=Q/C for capacitor
So DQ=-(Q/CR) Dt
DQ
1

Dt
Q
RC
dQ
1
or

Q
dt
RC
C
A
B
I
R
DQ
1

Dt
Q
RC
Q
t
1
1
Q Q dQ   RC 0 dt
0
1 t
ln Q   t 0
RC
t
ln Q  ln Q0  
RC
Q
Q0
Q  Q0 e
t

RC
Exponential
decay
Time constant, RC
1
When t=RC, Q  Q0 e
i.e. Q=0.368Q0
RC is known as the Time constant
Q=CV
It is the time taken for the charge (and V=IR
therefore also the voltage and current) to
drop to 1/e of its initial value
• In another period RC, the charge will have
dropped to 1/e2 of its initial value
• etc...
• cf radioactive decay constant
•
•
•
•
Discharging Capacitor
Capacitor Discharge
Summary:
Charging capacitors
[not examinable]
Capacitor Discharge
Other exponentials in nature
• Air leaking from a bike tyre
• Radiation absorption with distance
• Decay of earthquake wave amplitude with
distance
• Sand through an egg timer
• Population trends
• Moore’s law for semiconductors
• ...
Capacitors in parallel
•
•
•
•
Both have the same voltage V
QT=Q1+Q2
But Q=CV, so:
CTV=C1V+C2V, or
• CT=C1+C2
• …it’s just like increasing the size of the
plates…
Capacitors in series
• Both have the same charging current, so
each has the same displaced charge Q
• V=V1+V2
• V=Q/C, so
• Q/CT=Q/C1+Q/C2
• So
1
1
1


CT C1 C2