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for review about capacitor and capacitance
Typology: Lecture notes
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A capacitor is basically two parallel
conducting plates with air or insulating
material in between.
A capacitor doesn’t
have to look like
metal plates.
Capacitor for use in
high-performance
audio systems.
When a capacitor is connected to an external potential,
charges flow onto the plates and create a potential difference
between the plates.
Capacitor plates build up charge.
The battery in this circuit has some voltage V. We haven’t discussed what that means yet.
The symbol representing a capacitor in an
electric circuit looks like parallel plates.
Here’s the symbol for a battery, or an external
potential.
assortment of
capacitors
The magnitude of charge acquired by each plate of a capacitor
is Q=CV where C is the capacitance of the capacitor.
The unit of C is the farad but most capacitors have values
of C ranging from picofarads to microfarads (pF to F).
micro 10 -^6 , nano 10 -^9 , pico 10 -^12 (Know for exam!)
Here’s this V again. It is the potential difference provided by the “external potential.” For example, the voltage of a battery.
d
We previously calculated the electric field
between two parallel charged plates:
0 0
This is valid when the separation is small
compared with the plate dimensions.
We also showed that E and V are related:
d d
0 0
0
0
This lets us calculate C for
a parallel plate capacitor.
Reminders:
Q is the magnitude of the charge on either plate.
V is actually the magnitude of the potential difference
between the plates. V is really |V|. Your book calls it
Vab.
C is always positive.
We can also calculate the capacitance of a
cylindrical capacitor (made of coaxial
cylinders).
The next slide shows a cross-section view of
the cylinders.
Q
b r
a
E
d
Gaussian
surface
Q λ L λ L C = = = ΔV ΔV b 2k λ ln a
L^2 πε L 0 C = = b b 2k ln ln a a
Lowercase c is capacitance per unit length:
C 2 πε 0 c = = L b ln a
This derivation is sometimes needed
for homework problems!
b b
b a r a a
ΔV = V - V = - E d = - E dr
r r l
b
a
dr b ΔV = - 2k λ = - 2k λ ln r a
Let’s calculate the capacitance of a concentric spherical
capacitor of charge Q. I’ll skip this calculation if there is no related homework assigned.
In between the spheres
2 0
b
a^2 0 0
You need to do this derivation if you have a
problem on spherical capacitors!
+Q
b
a
Let aR and b to get the capacitance of an isolated
sphere.
+Q
b
a
alternative calculation of capacitance of isolated sphere
Example: what is the charge on each plate if the capacitor is
connected to a 12 volt* battery?
^
12
10
*Remember, it’s the potential difference that matters.
If you keep everything in SI (mks) units, the result is “automatically” in SI units.
Example: what is the electric field between the plates?
d = 0.
If you keep everything in SI (mks) units, the result is “automatically” in SI units.
Recall: this is the symbol representing a
capacitor in an electric circuit.
And this is the symbol for a battery… +-
…or this…
…or this.
Capacitors connected in parallel: C 1
C 2
C 3
+ (^) -
V
The potential difference (voltage drop) from a to b must equal V.
a b
Vab = V = voltage drop across each individual capacitor.
Vab
Note how I have introduced the idea that when circuit components are connected in parallel, then the voltage
drops across the components are all the same. You may use this fact in homework solutions.
