Goal: To understand the basics of capacitors

1
Goal To understand the basics of capacitors

• Objectives
• To learn about what capacitors are
• To learn about the Electric fields inside a
  capacitor
• To learn about Capacitance
• To understand how a Dielectric can make a better
  Capacitor
• To be able to calculate the Energy stored inside
  a capacitor

2
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.

3
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.

4
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 s
• s Q / A where Q is the total charge and A is
  the area
• And E 4p k s
• Also, E s / e0 where e0 is a constant (called
  the permittivity of free space)
• e0 8.85 10-12 C2/(Nm2)

5
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).

6
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?

7
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)

8
Finding the Capacitance of a Capacitor

• For this we have a few steps
• E s / e0
• Since s Q/A, E Q / (e0 A)
• V E d, so V Q d / (e0 A)
• Or, just moving things around
• Q/V e0 A / d
• Since C Q / V e0 A / d

9
Wake up time!

• 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.

10
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
• A) the Electric field between the plates.
• E s / (e0 )
• s Q / A, Q 5 C, and A 0.1 m 1 m 0.1 m2
• So, s (Done on Board)
• And E (Done on Board)

11
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 e0 / d (Done on Board)

12
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

13
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?

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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.

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Dielectrics

• The dielectric has the effect of increasing the
  capacitance.
• The capacitance is increased by a factor of the
  dielectric constant of the material (?).
• So, C ? A / (4p k d) or ? e0 A / d

16
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!

17
Energy

• Lightning of course contains a LOT of energy.
• So, clearly capacitors dont 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.

18
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?

19
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)

20
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