Capacitors

Capacitor

• Consists of 2 metal plates which are seperated by an insulating layer (known as a dielectric)

• It stores charge by building up a surplus of electrons on one plate and a deficit on the other

What can a dielectric (insulating layers) be made of?

• Air

• Plastic

• Paper

How do capacitors store charge?

• When a capacitor is uncharged it contains equal amounts of electrons on both plates

• When a capacitor is connected to a battery some of those electrons are moved from one plate to another

• The electrons will be attracted to the positive terminal of the battery and then repelled by the negative terminal

• The capacitor is now charged

Rules of a capacitor

• Can only be charged by a DC supply

• The capacitor cannot be charged to a higher voltage then the power supply

Capacitors vs batteries

• Stores energy as electrical potential energy (due to electrons being pushed together) - battery = chemical energy

• Capacitors store very small amounts of charge which can be released very quickly as a high current

Application of capacitors

Capacitance

The amount of charged stored per unit potential difference

Capacitance graph

• This assumes that current stays constant which is not true

Why does current not stay constant when charging a capacitor?

• There aren’t fixed spaces on a capacitor for electrons

• Instead as you pack the electrons closer together the force of repulsion between them increases

• Evenutally the force of repulsion is equal to the EMF (electromotive force) for the battery and so the current stops

How to keep current constant when charging a capacitor

• Use a variable resistor

• Set the resistance high to slow the flow of electrons in and gradually reduce

How is energy stored in a capacitor

• Capacitor energy is stored as electrical potential energy

• It’s achieved by forcing electrons closer together on one plate

• The more electrons that squeeze onto the plate, the smaller the distance between them, so these electrons will try to space out equally

• This increases the force between them (Coulomb’s law)

• Which takes energy to push the electrons onto the plate - the moment the power supply is taken away, they will burst back out

Relationship between energy stored and potential difference (voltage)

How does the energy stored in a capcitor compare to the energy stored in a battery

• Batteries produce a steady voltage

• In a capacitor half the energy gets lost as heat when moving electrons from one plate to the other

Dielectric

• The insulating layer between the two metal plates

• It prevents electrons from jumping from one plate to another

• Between the two parallel plates, a uniform electric field is generated

Relationship between permittivity and charge to generate an electric field

• Greater permittivity requices greater charge

• Directly proportional

Permittivity

How hard it is to generate an electric field in a material

Relative permittivity (Dielectric Constant)

• The ratio of the permittivity of a material compared to the permittivity of free space (a vacuum)

• Different dielectrics have different relative permittivity’s

Breakdown voltage

Refers to how strong the electrical field needs to be to allow electrons to ‘punch’ through the dielectric

Polar Molecules

• Permittivity can be explained by the motion of polar molecules inside a dielectric

• Polar molecules have a positive and negative end

• As the capacitor charges, one plate becomes positive and the other negative

• The positive ends will be attracted to the negative plate, and the negative ends to the positive plate

• Molecules are aligned anti-parallel to the field

• This means the charges on the field and molecules are the opposite way round

How relative permittivity affects capacitance

• Each polar molecule has its own electric field

• In this alignment, it opposes the field between the plates

• Permittivity (larger) is directly proportional to the opposing field (larger) - thereby creating a weaker overall electric field

• This makes it easier to continue transferring electrons from one plate to the other

• Therefore reducing the potential difference (Joules per coulomb) needed to transfer a given charge (electrons)

• So capacitance increases

Three factors that affect capcitance

• Area overlap (More area = More space to store electrons, C ∝ A)

• Permittivity of dielectric (Higher permittivity = makes it easier to transfer electrons C ∝ ε1 [ε0 x εr])

• Distance between plates (When distance is small the force between the plates is greater, allowing you hold more electrons in place, C ∝ 1/d

Capacitors in parallel

• When capacitors are placed in parallel you are increasing the area that electrons can be stored on

• Effectively creating one larger capacitor

• Which leads to an increase in capcitance, as C ∝ A

• C total = C1 + C2 + …

Capacitors in series

• When capacitors are placed in series you are effectively increasing the distance between the plates

• Which leads to a decrease in capacitance as C ∝ 1/d

• 1/C total = 1/C1 + 1/C2 + …

Capacitor charging graphs

• Current flowing into an uncharged capacitor starts off high and then falls exponentially

• The charge will build up quickly on an uncharged capacitor while current is high and then build up slower as current slows

• The voltage will do the same as charge as tjeu are dorectly proportional. It cannot go above the emf of the battery

How the voltage across the resistor varies in a charging circuit

• In a charging circuit, the capacitor and resistor will share the voltage produced by the battery

• But as the voltage across the capacitor increases, the resistor will get a decreasing share

• Until eventually the capacitor has a voltage to the emf, and the resistor will get none

Analogy for a discharging capacitor

• Use of coulomb’s law

• When a capacitor is charged, the electrons are close together

• Coulombs tells us that the smaller the distance the greater the force

• As the capacitor discharges and electrons leave the plate, the distance between them increases, which reduces the force

Capacitor discharging graphs

• Initial current from a charged capacitor will start high and then reduce exponentially over time

• The charge reduce quickly when current it high and then reduce more slowly as the current slows

• The voltage will do the same as charge as they are directly proportional

How voltage across the resistor varies in a discharging circuit

• In a discharging circuit the capacitor is the only power supply

• The resistor is the only other component

• So whatever voltage the capacitor is producing gets completely used by the resistor

• The voltage on both will be the same magnitude

• The reason why the voltage across the resistor is labelled as a negative is because it is flowing the opposite way to when the circuit was charging

Exponential decay equation for capacitors

• The graph is an exponential decay. This means we can predict the charge left at any point in time

• It also follows the half-life rule - this means the charge always takes the same amount of time to halve

Discharging equations

Charging equations

• The current graph is exponential decay whilst the others aren’t

What happens when a capacitor discharges into another capacitor

Following the circuit diagram:

• When the switch is at position 1, then capacitor C₁ is charged by the battery:

• This will cause capacitor C₁ to become fully charged and have a voltage of the emf

• This will happen almost instantly as the wires are the only thing providing resistance

• C₂ will not charge as it is part of a broken circuit

• When connected to position 2 then the fully charged C₁ discharges into C₂, charging it,

• C₁ is now the only power cupply, it will discharge into C₂, and charge it up

• Eventually C₁’s and C₂’s voltage will become the same

• The current will then stop flowing

• Remember that C₁ and C₂ are not the same capacitance, so the charge won’t be equal