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## Goal: To understand the basics of capacitors

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### 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

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

- 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!

- 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!

- 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 = κA / (4π k d) or κε0 * 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

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