Goal To understand the basics of capacitors Objectives

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 / ε 0 where ε 0 is a constant (called the permittivity of free space) • ε 0 = 8. 85 * 10 -12 C 2/(N*m 2)

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 / ε 0 Since σ = Q/A, E = Q / (ε 0 * A) V = E * d, so V = Q d / (ε 0 * A) Or, just moving things around: Q/V = ε 0 * A / d Since C = Q / V = ε 0 * 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 / (ε 0 ) σ = Q / A, Q = 5 C, and A = 0. 1 m * 1 m = 0. 1 m 2 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 ε 0 / 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 ε 0 * 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 V 2 • 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