A Level Physics Revision: All of Capacitors (in under 21 minutes)

Summary

This A-Level physics tutorial explores the fundamentals of capacitors, including their symbol, function, and the concept of capacitance. It explains how capacitors store electrical charge proportional to potential difference and introduces the formula for capacitance (C = Q/V). The tutorial covers charging and discharging processes, the role of electron flow, and how capacitors can serve as a power source. It also delves into series and parallel configurations, explaining the rules for calculating total capacitance in each case. The video further discusses the exponential nature of voltage and current changes during charging and discharging, introduces the time constant, and demonstrates how to derive it from graphs. Practical examples and calculations are provided to solidify understanding.

Takeaways

• 🔋 The symbol for a capacitor is two parallel lines, representing its function to store electrical charge.
• 🔌 Capacitors charge and discharge in a circuit, storing charge when connected to a power supply and releasing it when connected to a load like a light bulb.
• ⚡ The amount of charge a capacitor can store is proportional to the potential difference across it, with the constant of proportionality being the capacitance (C).
• ⚖️ Capacitance is defined as the amount of charge stored per unit of potential difference, measured in farads (F), where 1 F = 1 coulomb per volt.
• 🔄 When charging, electrons flow from the negative terminal of the power source to the negative plate of the capacitor, creating an equal and opposite charge on the positive plate.
• 🔌 In a parallel circuit with capacitors, the total capacitance (C_total) is the sum of individual capacitances (C1 + C2), and the voltage across all is the same.
• 🔗 In a series circuit with capacitors, the total capacitance is found using the reciprocal formula: 1/C_total = 1/C1 + 1/C2, and each capacitor experiences the same current.
• 📉 The voltage, current, and charge in a discharging capacitor decrease exponentially over time, following the formula V = V_initial * e^(-t/(CR)).
• 🕒 The time constant (τ) of a capacitor is the product of its capacitance and the resistance in the circuit, and it determines the rate of charge/discharge.
• 📈 The time constant can be determined from a graph of voltage, current, or charge over time, where the voltage drops to approximately 37% of its initial value after one time constant.
• 🔌 During charging, the voltage across a capacitor increases exponentially while the voltage across a resistor in series decreases, with the total voltage remaining constant due to Kirchhoff's second law.

Q & A

What is the symbol for a capacitor?

The symbol for a capacitor is two parallel lines.

What is the fundamental purpose of a capacitor?

The fundamental purpose of a capacitor is to store electrical charge.

How is the amount of charge stored on a capacitor related to the potential difference across it?

The amount of charge stored on a capacitor is proportional to the potential difference across it.

What is the constant of proportionality between the charge and potential difference on a capacitor called?

The constant of proportionality between the charge and potential difference on a capacitor is called capacitance, symbolized by 'C'.

What is the unit of capacitance and how is one farad defined?

The unit of capacitance is the farad. One farad is defined as the amount of capacitance when one coulomb of charge is stored for one volt.

How can the charging of a capacitor be explained in terms of electron flow?

During the charging of a capacitor, electrons flow from the negative terminal of the cell to the negative plate of the capacitor, causing excess electrons to gather on the negative plate and repelling electrons from the positive plate, leading to an equal and opposite charge on the plates.

How does the total capacitance change when capacitors are connected in parallel?

When capacitors are connected in parallel, the total capacitance is the algebraic sum of the individual capacitances, expressed as C_total = C1 + C2.

What is the rule for calculating the total capacitance when capacitors are connected in series?

For capacitors in series, the total capacitance C_total is calculated as the reciprocal of the sum of the reciprocals of the individual capacitances: 1/C_total = 1/C1 + 1/C2.

Why is the potential difference the same across all capacitors in a parallel circuit?

In a parallel circuit, the potential difference is the same across all branches because the voltage of the source (e.g., a cell) is distributed equally across all parallel paths.

How is the charge shared in a parallel circuit with capacitors?

In a parallel circuit, the total charge is shared among the capacitors, with the total charge Q_total being the sum of the individual charges Q1 + Q2.

What is the time constant in the context of capacitors, and how is it defined?

The time constant, denoted by the Greek letter tau (τ), is the product of the capacitance and resistance in a circuit. It represents the time it takes for the voltage (or current) to reach approximately 37% of its final value during the charging or discharging of a capacitor.

How does the voltage across a discharging capacitor decrease over time?

The voltage across a discharging capacitor decreases exponentially over time following the formula V = V_0 * e^(-t/(CR)), where V_0 is the initial voltage, t is the time, C is the capacitance, and R is the resistance.

What happens to the potential difference, charge, and current at time equals the time constant during capacitor discharge?

At time equals the time constant, the potential difference, charge, and current will drop to approximately 37% of their initial values during the discharge of a capacitor.

How can the capacitance of a capacitor be determined from a discharge graph?

The capacitance can be determined from a discharge graph by identifying the time constant from the graph, which is the time it takes for the voltage to drop to 37% of its initial value, and then dividing this time by the resistance in the circuit.

What is the relationship between the potential difference across a charging capacitor and the time since it started charging?

The potential difference across a charging capacitor increases with time, following the formula V_C = V_0 * (1 - e^(-t/(CR))), where V_0 is the initial potential difference, t is the time, C is the capacitance, and R is the resistance.

How does the current in a series circuit change as a capacitor charges?

As the capacitor charges in a series circuit, the current decreases exponentially over time, following the same formula as the potential difference across the resistor: I = I_0 * e^(-t/(CR)), where I_0 is the initial current.

Outlines

00:00

🔋 Introduction to Capacitors and Charging

This paragraph introduces the concept of capacitors in A-Level Physics, explaining their role in storing electrical charge. The symbol for a capacitor is depicted as two parallel lines. The paragraph details the process of charging a capacitor within a circuit, emphasizing the relationship between stored charge (q), potential difference (v), and the constant of proportionality, known as capacitance (C). The formula C = q/v is presented, along with the definition of capacitance as the charge stored per unit potential difference. The unit of capacitance, the farad, is also defined. The explanation of electron flow during charging is provided, describing how electrons move from the cell's negative terminal to the capacitor's negative plate, leading to the accumulation of charge and the development of an equal and opposite charge on the capacitor's positive plate. Additionally, the paragraph touches on the use of a capacitor as a power source in a circuit, explaining how a charged capacitor can power a light bulb and the resulting decrease in the capacitor's charge.

05:00

🔌 Adding Capacitors in Series and Parallel

This section delves into the principles of adding capacitors in both series and parallel configurations. For parallel connections, the total capacitance (C_total) is the sum of individual capacitances (C1 + C2), as demonstrated with two 300 microfarads capacitors totaling 600 microfarads. In contrast, for series connections, the total capacitance is calculated using the reciprocal formula: 1/C_total = 1/C1 + 1/C2, which is then inverted to find C_total. An example calculation for two 300 microfarads capacitors in series results in a total of 150 microfarads. The paragraph also discusses the misconceptions about capacitors in series and suggests a video in the playlist for further understanding. The rules for charging and discharging in series and parallel configurations are also highlighted, emphasizing that in parallel circuits, the voltage is the same across all components, while in series circuits, the current is the same through all components.

10:01

⚡ Understanding Capacitor Discharge and Exponential Decay

The paragraph discusses the process of discharging a capacitor through a resistor, illustrating the exponential decay of voltage over time. The voltage across the capacitor (v) is described by the formula v = v_naught * e^(-t/CR), where v_naught is the initial voltage, t is time, C is capacitance, and R is resistance. The paragraph provides an example calculation for a 500 microfarad capacitor discharging through a 500 kilo-ohm resistor, resulting in a voltage of 3.29 volts after 150 seconds. The concept of an exponential function is explained, emphasizing the constant property ratio in equal time intervals, indicating that the voltage decreases by the same factor over equal time periods. The paragraph also explains how to rearrange the exponential decay formula to solve for time (t) and introduces the concept of the time constant (τ), which is the product of capacitance and resistance and has units of time.

15:04

🕰 The Time Constant and Capacitor Charging

This section explains the significance of the time constant (τ) in the context of capacitor charging and discharging. The time constant is shown to be the time it takes for the voltage to drop to 37% of its original value, using the exponential decay formula. The paragraph demonstrates how to calculate the capacitance from a discharge graph, using the time constant and resistance values. It also discusses the charging process of a capacitor, where the voltage across the capacitor increases while the voltage across the resistor decreases exponentially. The relationship between the potential difference across the resistor (VR) and the capacitor (VC) is explained using Kirchhoff's second law, leading to the formula for the potential difference across the charging capacitor. The paragraph concludes with the formula for the current in the circuit during capacitor charging, which decreases as the capacitor charges due to the decreasing potential difference across the resistor.

20:05

🔚 Conclusion of Capacitor Concepts in A-Level Physics

The final paragraph wraps up the discussion on capacitors, summarizing the key points covered in the A-Level Physics specifications. It revisits the main topics, including the charging and discharging of capacitors, the calculation of capacitance, and the understanding of exponential decay. The paragraph encourages viewers to ask questions if they have any and thanks them for watching, indicating the end of the video script on capacitors.

Keywords

💡Capacitor

A capacitor is an electronic component that stores electrical energy in an electric field. It is composed of two conductive plates separated by an insulator. In the video, the concept of a capacitor is introduced as a device that stores electrical charge proportional to the potential difference across its plates, which is central to understanding its function in circuits.

💡Potential Difference (voltage)

Potential difference, or voltage, is the work needed per unit of charge to move a test charge between two points. It is a measure of electrical potential energy. In the script, the potential difference across a capacitor is what drives the charging process, and it is directly proportional to the amount of charge stored on the capacitor.

💡Capacitance

Capacitance is the measure of a capacitor's ability to store charge per unit voltage. It is the constant of proportionality between the charge and the voltage across the capacitor. The script explains that capacitance is defined as the amount of charge stored per unit potential difference, with the unit being the farad.

💡Electron Flow

Electron flow refers to the movement of electrons in a circuit, which is fundamental to electric current. The video describes how electrons flow from the negative terminal of a cell to the negative plate of a capacitor during charging, and how this movement results in the build-up of charge on the capacitor's plates.

💡Parallel Circuit

A parallel circuit is a configuration where multiple components are connected alongside one another, with each component having the same potential difference across it. The script explains that in a parallel circuit with capacitors, the total capacitance is the sum of the individual capacitances, and the voltage across each capacitor is the same.

💡Series Circuit

A series circuit is a configuration where components are connected end-to-end, with the same current flowing through each component. The video script discusses how in a series circuit with capacitors, the total capacitance is calculated differently, with the reciprocal of the total capacitance being the sum of the reciprocals of the individual capacitances.

💡Charge Sharing

In a parallel circuit, charge sharing refers to the distribution of total charge among the capacitors. The script mentions that in practice, the total charge in a parallel circuit is the sum of the charges on each capacitor, which is a result of the shared potential difference.

💡Exponential Decay

Exponential decay describes a process in which the value of a quantity decreases exponentially over time. The video explains that when a capacitor discharges through a resistor, the voltage across the capacitor decreases exponentially, following a specific formula that includes the initial voltage, capacitance, and resistance.

💡Time Constant

The time constant, denoted by the Greek letter tau (τ), is the product of capacitance and resistance in a circuit. It represents the time it takes for the voltage across a capacitor to charge or discharge to approximately 37% of its final value. The script illustrates how the time constant is used to calculate the voltage across a capacitor at a given time after it begins to discharge.

💡Discharging

Discharging refers to the process where a charged capacitor releases its stored energy into a circuit. The video script describes the exponential decay of voltage across a capacitor as it discharges through a resistor, which is a key concept in understanding the behavior of capacitors in circuits.

💡Charging

Charging is the process by which a capacitor accumulates an electric charge, typically from a power source. The script explains the charging process of a capacitor in a circuit with a resistor and an EMF source, detailing how the voltage across the capacitor increases over time as it charges.

Highlights

Introduction to capacitors and their symbol representation in the A-Level Physics curriculum.

Explanation of a capacitor's function to store electrical charge proportional to the potential difference.

Definition of capacitance as the amount of charge stored per unit of potential difference.

The unit of capacitance, the farad, defined in terms of charge and voltage.

Describing the charging process of a capacitor in terms of electron flow.

Use of a capacitor as a power source in a circuit with a light bulb.

Calculating total capacitance in parallel circuits by simple addition.

Deriving the formula for total capacitance in series circuits and its relation to individual capacitors.

Common misunderstandings about capacitors in series and parallel circuits clarified.

The relationship between voltage, charge, and current in parallel circuits.

The counterintuitive nature of charge distribution in series circuits.

Discharging a capacitor through a resistor and the resulting exponential decrease in voltage.

The formula for voltage, charge, and current during capacitor discharge.

Application of the exponential decay formula to a practical problem involving a 500 microfarad capacitor.

The concept of an exponential function and its constant property ratio.

Rearranging the exponential equation to solve for time.

Introduction and explanation of the time constant in capacitor charging and discharging.

Calculating the time constant from a graph of voltage against time.

The charging process of a capacitor in a circuit with an EMF source and resistor.

The relationship between the potential difference across a charging capacitor and the resistor.

Current decrease in a series circuit as a capacitor charges.

Conclusion summarizing the capacitor portion of the A-Level Physics specifications.

Transcripts

hello physicists today we are going to

be looking

at all of the capacitors section on the

a-level physics

specification well let's get started

first of all the symbol for for a

capacitor is as

follows it is two parallel lines

now let's imagine that we put a

capacitor

and a little circuit in which we have a

cell

this capacitor here is going to start

charging

the idea of a capacitor is to store

electrical charge the amount of

charge that is stored on the capacitor

will be proportional to the potential

difference

v across the capacitor

and across the power supply that is

charging it

the constant of proportionality between

them

is known as the capacitance

and it is given the symbol

c rearranging for c

we get that c is equal

to q divided by v

and this leads us to our definition of

capacitance that

capacitance is the amount of charge

stored per unit of potential

difference the unit for capacitance is

known as the

ferret one farad is defined

as the amount of capacitance we would

get

when we have one coulomb of charge

that has been stored for one

volt can explain the

charging of a capacitor in terms of

electron flow

first off in a cell during the

charging of a capacitor we're going to

just connect this cell

to a capacitor as we can see over here

the electrons are going to flow from the

negative terminal of the cell

to the negative plate of

the capacitor there's going to be some

excess electrons that are going to

gather on the negative plate

and those are going to repel some

electrons

of the positive plate of the capacitor

which are going to move towards the

positive terminal of the cell overall

the plates are going to acquire an

equal and opposite charge

on the other hand we can also use a

capacitor

essentially as as a source of

power in a circuit for instance we can

take

a charged capacitor and we can connect

it to a

light bulb when we connect the capacitor

to this circuit the excess electrons are

going to flow

from the negative plate in this case

this will be clockwise to the

positive plate and the charge on the

capacitor

is going to be decreasing

let's go over adding capacitors in

series and parallel we're going to start

off with adding capacitors

in parallel first so if we have

capacitors in power

parallel for instance here we have two

capacitors of 300 microfarads which have

been connected in parallel you can see

the junctions

over here in parallel the total

capacitance i'm going to call that

c total c total

is equal to just the algebraic sum of

the two numbers

so that would be let's say c1 plus c2

applied to this problem above the total

capacitance

is going to simply be equal to

300 plus 300 micro farads so let's write

that 300 plus 300 which will be

equal to 600

microfarads

if the two capacitors are connected in

series the rule is as follows

1 over the total capacitance c total

will be equal to

1 over c1 plus 1 over

c2 it is best to input this into a

calculator by just finding the c

total as brackets one over c

one plus one over c two

then raise this whole expression to a

power of minus one

for instance the total capacitance in

this case will be equal to

one over three hundred plus

one over three hundred raise this whole

expression to a power of minus one

which is going to give us a hundred and

fifty micro farads

if you're wondering how to derive those

two rules or why they're exactly the

opposite to adding resistors

uh check out my video in the playlist of

my full online lessons

on capacitors which is in the

description

here are a couple of more commonly

misunderstood

facts about capacitors in a series

and in a parallel circuit let's start

off with

a parallel circuit first off because

this is a parallel circuit

the voltage or the potential difference

is exactly the same

across all the different parallel

branches

so the potential difference of our

source of emf in this case this is a

cell

it's going to be the same as the

potential difference over the

first capacitor when fully charged and

the potential difference

over the second capacitor when fully

charged

however because the current is actually

shared in a parallel circuit is being

added up

this also means that the charge

is going to be shared as well in

practice

this means that the total charge

q total will be equal to q1

plus q2 i mean if we wanted to

we could also write that c

total the total capacitance times

the potential difference will be equal

to c1

times v1 plus c2

times v2 however because the

potential difference is exactly the same

the voltage will be also

exactly the same and this is why we can

also see

that the rule for capacitance is that c

total is c1 plus c2

because all of the potential difference

is going to

be cancelled out

in a series circuit each capacitor

is going to experience the same current

so

this means that it will acquire the same

amount of charge

and this is really really important

because it's really counterintuitive

this will also be the total amount of

charge within the

circuit so q total will actually be

equal to q1 which will actually be equal

to q2 the pd

is shared by kirchoff's second law

and which says that the sum of the emf

so let's call that

v total will be equal to v1

plus v2 let's have a look at

a capacitor discharging for instance we

have this capacitor here that is

connected

across a resistor we're measuring the

voltage across this resistor and if we

did that

we are going to see that it will be

decreasing

exponentially the equation for the

voltage across a capacitor at a time

t after it begins to discharge that v is

equal to v

naught e to the minus t over c

r where v naught is the initial voltage

t is the time of interest c is the

capacitance and

r is the value of this resistor

because q is equal to cv and c is

constant this equation also applies for

the

charge so q is equal to q naught e to

the minus t over c

r additionally it will also apply for

the current which will also be

decreasing exponentially applied this

equation to a little problem over here

we have a 500 microfarad capacitor which

is charged up to

6 volts which is going to be our initial

voltage it is then discharged for a

500 kilo ohm resistor the value of the

resistance is quite large so i'm

expecting

the voltage to still be relatively high

and find the pd across the capacitor 150

seconds after it begins to

discharge okay well let's write down my

equation over here so i'm going to say

that

v is equal to v naught e to the

minus t divided by c r my initial

voltage v

naught is 6.0 volts

i'm going to multiply this by the

exponential of minus t

which is 150 seconds i'm going to divide

this

by cr which is 500

micro farad so it's time 10 to the power

of -6 farads

times my resistance which is 500 kilo

ohms that's 500 times

10 to a power of 3. when we carefully

input this into a scientific calculator

we're going to get 3.29

volts at time is equal to being equal to

150 seconds

after the beginning of the discharge now

let's have a look at what

actually makes a function an exponential

function

we need to know the constant property

ratio of

exponential in equal time intervals

the initial voltage over another one

let's call it v1 is going to be equal to

v1 divided by v2 is going to be equal to

v2

divided by v3 in practice this

means that in equal time intervals the

exponential

function will be decreasing by the same

factor for instance let's say that the

value decreases by a factor of a third

from v1 to v2

this would also indicate that the

function will also

decrease by another third from v2

to v3 this is the nature of the

exponential decay talking about

exponential functions

let's also remember how to rearrange

this equation

let's say for the amount of time

t so i'm just going to write over here

for practice we can

rearrange for

t the first thing that we're going to do

will be to

just rearrange for the exponential so v

over v naught will be equal to e to the

minus t over c r then i'm going to

take the natural log of both sides of

this equation

so i'm going to get that ln of v over v

naught is equal to the ln

of e to the minus t over cr

now remember the natural log and the

exponential functions

are inverse functions which means that

essentially they undo each other

so this would mean that ln of v over

v naught well that's not v okay there we

go

v over v naught will be equal to minus t

over c r

then what i'm going to do is i'm going

to multiply both functions

by the power of -1

so what i'm gonna get is that minus

ln of v over v naught is equal to

t over c r

the minus sign is actually going to

flip the natural log so

this would mean that this will be ln of

v naught

over v which is equal to t

over c r and finally all i need to do is

just rearrange

for the time i'm going to get that t

is equal to c r multiplied by the

natural log

of v naught over v

let's revise this property known as the

time constant next the charging and the

discharging of a capacitor depends

only on the resistance and the

capacitance

so we define this quantity which is

normally given the greek letter

tau as capacitance times resistance

this is actually a time let's show that

it has the

units of time so i'm just going to

multiply the two quantities

so c times r is going to equal

now because q is equal to

just going to write this on the side q

is equal to cv so this means the

capacitance is

q over v so i'm going to put that in

there

this will be q over v times resistance

which you

which is just v over i like so

those guys are going to cancel out

which means that i'm left with q

over i but remember q is equal to

i times t which means that time

is equal to q over i so this quantity

c times r actually has units of

time so this

time constant t is going to have the

units of time which

are seconds

what happens at time is equal to the

time constant this seems to be a special

time

so in fact let's calculate the

pd at time is equal to the time constant

so i'm just going to write down my

exponential equation v is equal to

v naught e to the minus t over

c r however rather than just time

i'm going to write the time constant

well let's see what happens v is equal

to

v naught e to the minus

now the time constant is just cr

so what i'm going to get is cr divided

by

cr and as we can see no pun intended

those two guys are going to cancel out

which will lead us

to the fact that v is v naught

e to a power of minus one

now e to a power of minus one we could

just put this

into a calculator and what we're going

to get

is that v will be 0.37 up to two

significant figures

times the original voltage v naught and

in fact this is the

very definition of the time constant

that a time equal to a time constant the

pd

will drop to 37 of its original value

so will the charge and so will the

current we can often figure out the time

constant directly

from a graph this is quite an obvious

example but the principles of it can be

applied to

most graphs of potential difference or

current or charge against time

for instance we have the exponential

decrease in pd

in a circuit in which a capacitor is

discharging for a five

kilo ohm resistor we can use this

graph to figure out the capacitance of

the capacitor first off if we have this

graph

we have the time constant because

the voltage is initially 100 then drops

down to

37 volts which is 37 percent of its

initial value

in 25 seconds this would mean that the

time constant t

is equal to 25 seconds

however remember that the time constant

is the product of the capacitance

times the resistance which would mean

that our capacitance

will be equal to our time constant

divided by the resistance

which will be 25 divided by 5 kilo ohm

resistors going to be 5 times 10 to the

power of 3

and if we put that into a calculator

this will

just give us five times

ten to a power of minus three farads

finally let's have a look at the

charging of a capacitor in order to do

so let's imagine a

circuit of emf v node that we've

connected to a resistor r

and a capacitor which is initially not

charged as soon as this circuit is

connected

this capacitor will start charging

now as the capacitor starts charging

the voltage across the capacitor will

increase

so this guy over here will start

increasing

and the voltage across the resistor

will start decreasing exponentially

in fact the pd across the resistor will

be decreasing

exponentially so vr will be equal to

v naught e to the power of minus t

over c r you can kind of think of it as

as

soon as this circuit is switched on vr

is equal to v naught and this guy is

equal to zero

and then vc increases while vr

decreases exponentially we can use

kirchhoff's

second law the sum of the emfs in this

case we only have one emf which is

v naught the potential difference across

the cell

will be equal to v r

plus v c which is the voltage across the

capacitor

and what we can do is just

write down that v naught is random vr

i'm gonna write v naught e to the minus

t

over c r plus vc

i can rearrange for the pd across the

capacitor

and i'm going to get that vc will be

equal to

v naught minus v naught e to the

minus t over c r we can take a factor of

v naught and what we're going to get is

that the potential difference across

the charging capacitor will be equal to

v

naught 1 minus

e to the minus t over

c r like so which is a formula for the

pd across this

capacitor like so the formula for the pd

across the resistor is given

here the same formula

also applies for the charge across the

capacitor

because q is equal to cv so what we can

write over here is that

q across the capacitor equal to

q naught 1 minus e to the minus

t over c r this will give us the charge

across the capacitor

at a time t however if we wanted to find

the

current in this circuit notice that the

current

will be dropping down in this series

circuit

as the capacitor is charging this is

because the pd

across the resistor is going down so

what i'm going to say

is that the current in the circuit will

be decreasing

as the capacitor

is charging

and will be decreasing by the same

factor

as the potential difference across the

resistor

and in fact in this circuit i will be

equal to

the initial current which will be the

highest e to the

minus t over c r when the capacitor

has stopped charging and has reached the

maximum potential difference then the

current will be

dropped to zero okay folks well this

was the vast majority of the

capacitor portion of the a-level physics

specifications

hopefully you have found this useful if

there are any questions please leave a

comment

and thank you very much for watching