Transcript Goal: To understand the basics of capacitors

Goal: To understand the
basics of capacitors
Objectives:
1) To learn about what capacitors are
2) To learn about the Electric fields inside a
capacitor
3) To learn about Capacitance
4) To understand how a Dielectric can make
a better Capacitor
5) To be able to calculate the Energy stored
inside a capacitor
What are capacitors?
• Much like we build reservoirs to hold water
you can build a device which holds onto
charge.
• These are capacitors.
• They work by separating + and – charges
so that you have an electric field between
them.
• Most commonly this is done on a pair of
plates which are parallel to each other.
Electric field inside a capacitor
• The electric field is usually a constant
between the plates of the capacitor.
• This makes the math fairly straight
forward.
• The voltage across the capacitor is
therefore V = E d where d is the
separation between the plates.
• Now we just need to find E.
Electric Field
• Each plate will have some amount of charge
spread out over some area.
• This creates a density of charge which is
denoted by the symbol σ
• σ = Q / A where Q is the total charge and A is the
area
• And E = 4π k σ
• Also, E = σ / ε0 where ε0 is a constant (called the
permittivity of free space)
• ε0 = 8.85 * 10-12 C2/(N*m2)
Capacitance
• Capacitance is a measure of how much charge
you can store based on an electrical potential
difference.
• Basically it is a measure of how effectively you
can store charge.
• The equation is:
• Q = C V where Q is the charge, C is the
capacitance (not to be confused with units of
charge), and V is the voltage (not to be confused
with a velocity)
• C is in units of Farads (F).
Quick question
• You have a 10 F capacitor hooked up to a
8 V battery. What is the maximum charge
that you can hold on the capacitor?
Quick question
• You have a 10 F capacitor hooked up to a
8 V battery. What is the maximum charge
that you can hold on the capacitor?
• Q = C V = (to be done on board)
Finding the Capacitance of a
Capacitor
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For this we have a few steps:
E = σ / ε0
Since σ = Q/A, E = Q / (ε0 * A)
V = E * d, so V = Q d / (ε0 * A)
Or, just moving things around:
Q/V = ε0 * A / d
Since C = Q / V = ε0 * A / d
Wake up time!
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Sample problem.
Two parallel plates are separated by 0.01 m.
The plates are 0.1 m wide and 1 m long.
If you add 5 C of charge to this plate then find:
A) the Electric field between the plates.
B) The Capacitance of the plate.
C) The voltage across the 2 plates.
Wake up time!
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Two parallel plates are separated by 0.01 m.
The plates are 0.1 m wide and 1 m long.
If you add 5 C of charge to this plate then find:
A) the Electric field between the plates.
E = σ / (ε0 )
σ = Q / A, Q = 5 C, and A = 0.1 m * 1 m = 0.1 m2
So, σ = (Done on Board)
And E = (Done on Board)
Wake up time!
• Two parallel plates are separated by 0.01
m.
• The plates are 0.1 m wide and 1 m long.
• If you add 5 C of charge to this plate then
find:
• B) The Capacitance of the plate.
• C = A ε0 / d = (Done on Board)
Wake up time!
• Two parallel plates are separated by 0.01
m.
• The plates are 0.1 m wide and 1 m long.
• If you add 5 C of charge to this plate then
find:
• C) The voltage across the 2 plates.
• V = Q / C or E * d
• Lets use E * d
Limits
• There are limits to what you can do with a
normal capacitor (just like limits to what
you can do with a dam).
• Eventually the charges will overflow the
capacitor and will leak out.
• How would you solve this problem?
Fill it with substance
• One solution is to place a material in
between the plates which prohibit the flow
of charge (an insulator).
• This allows you to build up more charge.
• A substance that allows you to do this is
called a dielectric.
Dielectrics
• The dielectric has the effect of increasing
the capacitance.
• The capacitance is increased by a factor
of the dielectric constant of the material
(κ).
• So, C = κ A / (4π k d) or κ ε0 * A / d
Lightning!
• One natural example of a discharging capacitor
is lightning.
• Somehow the + charges are removed from the –
ones in the updraft of the cloud.
• So, the bottom of the cloud has – charge.
• This induces a + charge on the ground.
• Now they do a dance. The – charges step down
randomly. The + charges step up randomly.
• If they meet it forms a pathway for a large
amount of charge to flow very quickly – a
lightning strike!
Energy
• Lightning of course contains a LOT of energy.
• So, clearly capacitors don’t just keep charge, but
energy as well.
• How much energy?
• For a plate capacitor the energy it stores is
simply:
• U = ½ Q V or ½ Q E d or ½ C V2
• Note this is half of what we had for individual
charges – be careful not to mix up the equations
for particles and capacitors.
Sample
• You hook up a small capacitor to an 8 volt
battery.
• If the charge on the plates are 5 C then
how much energy does the capacitor
contain?
Sample
• You hook up a small capacitor to an 8 volt
battery.
• If the charge on the plates are 5 C then
how much energy does the capacitor
contain?
• U = ½ Q V = (Done on Board)
conclusion
• We learn that capacitors act as dams for
charge – allowing them to store charge.
• Store too much though, and they flood.
• The maximum charge storable is Q = VC
• Dielectrics can increase this by increasing
the capacitance.
• We learn the equations for capacitance
and the E field inside a capacitor.
• The energy a capacitor holds is U = ½ Q V