chapter 24: capacitance and...

Chapter 24: Capacitance and Dielectrics• When you compress/stretch a spring, we are storing potential energy … This is the mechanical method to store energyIt is also possible to store electric energy as electric potential energy using capacitors

• In this chapter: we will study electric devices (capacitors) used to store electric energy

‐We Will Learn The Followings‐• The nature of capacitors (how we can make them) and their ability to store energy There are various types of capacitors, we will define the physical quantity (capacitance) which is a measure of ability to store energy and learn how to calculate this quantity for various capacitor like the – plate type – cylindrical – spherical …

• Analysis of capacitors connected in a network (connected in parallel and series)There might be more than one capacitor connected in a circuit …

• Calculation of energy stored in a capacitor (how to calculate the stored energy)We will drive an expression that gives the total amount of stored energy in a capacitor.

• Study of dielectrics in more details and how they used to make capacitors more effectiveWe will learn how we can use the dielectrics to increase the efficiency of capacitors.

Capacitors and Capacitance• Capacitors are the devices that store electrical energy as electric potential energy• A capacitor can be made using two conductors insulated from each other

Q = 0 (uncharged – empty)no charge means the capacitors are not charged or empty

Q ≠ 0 (charged)One can charge this capacitor by using a battery by applying a potential difference – which will reveal +Q and –Q on each conductor. This process is called charging the capacitor.The battery doesn’t produce any charged. It just does move the charges from one conductor to another through chemical process until the potential difference becomes equal to potential difference along the battery.

The amount of charge on each conductor Q is proportional to potential difference between the conductors. This proportionality constant / ratio between Q and Vab is called capacitance: Capacitors in Circuit Diagrams

Capacitance *** Q ~ Vab ***proportionality constant is called capacitance, C .

When potential difference increases Q increases, but C remains the same.

C is a CONSTANT that ONLY depends on the physical properties of the capacitor ‐ mostly the geometry

C is a measure of ability to store energy. Greater Cmeans that it can hold/store more charge and energy

Unit for capacitanceThe SI‐unit of capacitance is called one farad (1 F), named after 19th century English physicist Michael Faraday.

Attention: Don’t confuse C with CC is used for coulomb (unit for charge) but C (italic C) is for capacitance.

Calculating Capacitance: Capacitors in VacuumThere are various type of capacitors commonly used in applicationsParallel‐plate Capacitors – formed using two conducting parallel platesSpherical Capacitors ‐ formed using two conducting spherical shellsCylindrical Capacitors ‐ formed using two conducting cylindrical shells

Calculation of Capacitance for these capacitors is simple.For a capacitor, we can calculate the capacitance by considering a charge Q then calculate the potential difference between them and then applying formula that describes the capacitance. Here, we will consider the capacitors in vacuum where the space between the conductors are empty. Later we will learn using the dielectrics to increase the capacitance (measure of efficiency in storing energy) of a capacitor.

Parallel‐Plate Capacitors ‐ CalculationWhat is the capacitance of a capacitor which is formed using two parallel plates?The separation between the plates is d and surface area for the plates is A.

In order to determine the capacitanceConsider plates are charged with +Q and –QDetermine the potential between + and – platesApply capacitance formula C = Q/Vab

If the d << size of the plates then the field is uniform directed from + to – plate.

From Gauss’s Law: which can be expressed in terms of Q:

Potential difference: yields to

Now, Apply capacitance Formula: C = Q /Vab

00

Q Q AC CEd Q A d d

Parallel‐Plate Capacitors ‐ InterpretationAs mentioned before, capacitance of any capacitor only depends on physical properties of the plate and vacuum (that fills the space between the pates)

0ACd

‐ Capacitance ‐Increases with increasing ADecreases with increasing d

Capacitance also depends on ε0 (electric permittivity of free space) which is the nature of space between the plates. Later, we will see that we can use some dielectric materials to increase the capacitance instead of using vacuum.

Let’s now do a unit analysis for the expression that we have found.As you remember from Coulomb’s law, unit for ε0: plug this into eq.

Remember Joule/Coulomb = Volt – So F = Coulomb/Volt is dimensionally correctTherefore, we can take the unit for ε0 alternatively in capacitance calculations

Note: Because ε0 is very small, capacitors in industry are usually in microfarads or smaller, like picofarad 1pF = 10‐12 F

Various Types of Commercial Capacitors Capacitance and Break–Down Voltageare marked on each capacitor

As can be seen in this pictureCapacitance increases with increasing size

Breakdown voltage (6.3V) depends on thematerial used in construction, usually the electric properties of the material between the plates

Capacitors in Series and ParallelCapacitors can be found in industry might not be the one what you actually need.

Usually manufacturers makes the capacitors with certain capacitance and working voltages.One may always use a combination of capacitors to get what is need.

There are two possible combinations in connecting two capacitors

1) Series connection :capacitors are connected one after another

2) Parallel Connection: capacitors legs are connected side by side

For both cases:The connected capacitors acts like one capacitor with a capacitance called “equivalent” capacitance.

Capacitors in SeriesWhen two capacitors are connected in series and a potential difference Vab=Vapplied across the capacitors, Q charge will be charged on both capacitors.Because the plates in between just exchange charges

Also the potential across a and b must be the sum of ac and cbLet’s call these potentials as V, V1 , and V2 1 2V V V

1 21

1 22

By using C= for and and : Q C C Q QVV

VC C

eqeq

By using C= for : QVCC

QV

1 2Potential differences must satisfy: V V V

eq 1 2 eq 1 2

1 1 1= = Q Q QC C C C C C

Reciprocal of the equivalent capacitance equals to sum of reciprocals of the individual capacitances

Capacitors in ParallelWhen two capacitors are connected in parallel and a potential difference Vab=Vapplied across the capacitors, same potential difference exist along each capacitors that will cause charging Q1 and Q2 on each capacitors.

Net/total charge charged : 1 2Q Q Q

1 2 1 1 2 2By using C= for and and : Q C C Q C QV

V C V

eq eqBy using C= for : QC CQV

V

1 2Total charge must satisfy: Q Q Q Q

eq 1 2 eq 1 2 + +C V CV C V C C C

The equivalent capacitance equals to sum of the individual capacitances

Q charge is stored on the capacitors – net Q not 2Q

Where

and “equivalent capacitance”

Potential Differences: and

Combinations of Capacitors – Comparison ‐1) Series Connection :

2) Parallel Connection: V potential difference across each capacitor is same V=V1=V2

Stored charges on each capacitor: Q1=C1V and Q2=C2V

and “equivalent capacitance”

Net/Total charged stored: Q = CeqV = Q1 + Q2

eqQ C V

In the network some of them is parallel and some of them are in series, we can reduce the network by applying the equivalent capacitance idea until we get one capacitance.

Energy Stored in Capacitors – Derivation‐So far, we have called the capacitors as the electric devices used to store electric energy.Now, lets try to calculate an expression for the energy stored in a capacitor by considering the simplest parallel‐plate capacitor, and keep in mind that this expression will also be valid for any type of capacitor .

Consider the plates are initially uncharged Q=0 and the batter will do some work to move the charges from one plate to other plate until the charge becomes Q=Q.

As we know that the V potential difference across a capacitor increases while the Q increases, so the work needs to be done will be increasing to bring extra charge. For this reason, we need to use an integral expression to calculate the work. This integral, will gives us the total work needs to be done to charge the capacitor with a charge Q, also equals to potential energy stored in the capacitor. Usually this is the work done by the battery.

Now, lets say at any time, charge on the capacitor is q and a dW work needs to be done to bring extra dq onto q charge. When there is dq charge on capacitor v (pot. diff) = dq/C.

Energy Stored in Capacitors – Electric‐Field EnergyThe expression for the potential energy for a charged capacitor:

where C in Farads, V in Volts, and Q in Coulombs U in joule

This expression can be used to calculate the energy stored in any type of capacitor.

Energy Stored in Electric Field: Electric‐Field EnergyEnergy stored in a capacitor can be used to define the energy stored per unit volume by means of electric‐field. In order to keep it simple let’s consider a parallel‐plate capacitor charged with charge Q.+Q ‐Q

00 0

/ and and Q A AE C V Edd

Substitute C and V into above equation and after arranging it:

Ad is the volume between the plates:

20

1 ( )2

U E Ad

20

1 2

u E Energy Stored by E. Fieldper unit volume

Dielectrics – nonconducting materials‐Space between the conductors inside a the capacitors are mostly filled with dielectric materials.

***This process mainly serves three functions***1) Helps to maintain the conductors in a small separation without contact.2) It increases the breakdown value for the capacitor (max voltage it can hold).3) Increases the capacitance of capacitor comparing to formed in vacuum.

Let’s try to understand the 2nd and 3rd functions of the dielectric in a capacitorConsider a charged capacitor then insert a dielectric material between the conductors. A voltmeter is connected along the capacitor to keep track of the change in the potential.

Initially: Q and V0C0 = Q/V0

Finally: Q and V where V < V0C=Q/V

Because V < V0 then C > C0 .

Potential decreases with the dielectric.Capacitance increaseswith the dielectric.

Dielectrics – Dielectric ConstantThe capacitor filled with dielectric increases, using this idea “dielectric constant”, K is defined as a measure of this increase comparing to C in the vacuum.

0 0where is the capacitance in the vacuum.C KC C

0

called dielectric constant. ( )CK no unit unitless quantityC

Page 801

K=1 for vacuum and something greater than 1 for other dielectrics.

0

i 0

solving for along with the above equations for and

EEK

E E

Induced Charge and PolarizationLet’s now try to explain why the potential difference gets smaller when a dielectric is inserted into a capacitor, which is eventually results an increase in capacitance

The experiment has revealed that the potential difference across the plates in a capacitor decreases. If we remember that E = V/d then the electric field is also expected to be decreases. Let’s analysis the field between the parallel plates.

00 where is the electric field in the vacuumEE E

K

The electric field gets smaller because of the polarization (charge separation) effect in the dielectric material. Where σi represents the induced charged density on the dielectric. We can determine this σi in terms of K dielectric constant. Using Gauss’s Law E= σnet /ε0

Induced charge density:

Electric Field inside a Dielectric0

00

0

inside the dielectric electric field: where

which can be written as

EE EK

EK

Recall that ε0 is the “permittivity of the vacuum/free space”.

ε =Kε0 quantity is called as the “permittivity of the dielectric”.

In terms of , the electric field inside ...

E

So, you can use any previous equation by replacing ε0 with ε

or in the energy expression of the electric field

Dielectric BreakdownWhen we put a dielectric in an external electric field, it causes polarization.If the electric field gets bigger, the polarization (charge separation) gets bigger.Once the electric field gets enough large then the electrons inside dielectric starts to become ripped and dielectric becomes like conductor (charges start to flow through the dielectric just like it is a conductor).

This maximum electric field that a dielectric can sustain/tolerate is called “Electric Strength” of the dielectric material. For dry air/vacuum ~ 3x106 V/m .

Gauss’s Law inside a DielectricWhen we are studying electric field inside a dielectric and if we need to use Gauss’s Law then we can use the Gauss’s Law in the following form

Where K is the dielectric constant and Qencl‐free accounts for the free charges, not the induced charges. Then E becomes the electric field inside the dielectric.

Or we may just replace ε0 with ε and use the following formulaRecall that ε=K ε0

encl-freeQ =E dA

EXAMPLE: Electric Field between parallel plates – filled with a dielectric, K

Considering the closed integral through cylindrical (like a can) Gauss surface

AEA

0

EK

Space between conductors is filled withdielectric material with dielectric constant K

E between the plates = ?

1,3,4,5,6,7,8,9,12,14,15, and 17

1,2,3,4,5,6,8,9,10,11,12,13,14,16,17,18,19,20,21,22,23,24,25,26,30,31,33,35,36,37,41,43,45, and 46

RECOMMENDED END OF CHAPTER‐24QUESTIONS AND PROBLEMS

53,54,55,57,64,65,66,68, and 72

chapter 24: capacitance and...

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