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General Physics (PHY 2140)

Lecture 3Lecture 3Electrostatics

Electrical energypotential difference and electric potential

potential energy of charged conductors

Capacitance and capacitors

Chapter 16

http://www.physics.wayne.edu/~alan/

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Lightning ReviewLightning Review

Last lecture:

1.1.

Flux. GaussFlux. Gauss’’s law.s law.simplifies computation of electric fieldssimplifies computation of electric fields

2.2.

Potential and potential energyPotential and potential energyelectrostatic force is conservativeelectrostatic force is conservativepotential (a scalar) can be introduced as potential potential (a scalar) can be introduced as potential energy of electrostatic field per unit chargeenergy of electrostatic field per unit charge

Review Problem:

Perhaps you have noticed sudden gushes of rain or hail moments after lightning strokes in thunderstorms. Is there any connection between the gush and the stroke or thunder? Or is this just a coincidence?

cosneto

QEA θε

Φ = =∑

B APEV V Vq

ΔΔ = − =

Emg

CF qE=

cloudV

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Electrostatic PrecipitationElectrostatic Precipitation

Gushes of Rain and Hail After LightningGushes of Rain and Hail After Lightning IssnIssn:: 15201520--0469 0469 Journal:Journal: Journal of the Atmospheric Sciences Journal of the Atmospheric Sciences Volume:Volume: 21 21 Issue:Issue: 6 6 Pages:Pages: 646646--665665

Authors:Authors: Moore,Moore,

C.B., Vonnegut,C.B., Vonnegut,

B., B., VrablikVrablik,,

E.A., E.A., McCaigMcCaig,,

D.A.D.A.

Article ID:Article ID:10.1175/152010.1175/1520--0469(1964)021<0646:GORAMA>2.0.CO;20469(1964)021<0646:GORAMA>2.0.CO;2

ABSTRACTABSTRACTObservations of thunderstorms in New Mexico were made with a veObservations of thunderstorms in New Mexico were made with a verticallyrtically--scanning, scanning, 33--cm radar on a mountain top. Prior to a cloudcm radar on a mountain top. Prior to a cloud--toto--ground lightning discharge nearby, ground lightning discharge nearby, the radar echo overhead was usually quite weak, indicating low ithe radar echo overhead was usually quite weak, indicating low intensities of ntensities of precipitation there. Following the lightning it was observed somprecipitation there. Following the lightning it was observed sometimes that in the etimes that in the region of the cloud where the discharge occurred the radar echo region of the cloud where the discharge occurred the radar echo intensity rapidly intensity rapidly increased, and shortly thereafter a gush of rain or hail fell neincreased, and shortly thereafter a gush of rain or hail fell nearby. arby. These studies confirm earlier radar observations, made by the aThese studies confirm earlier radar observations, made by the authors at Grand uthors at Grand BahamaBahama

Island, B.W.I., in which it was found that lightning is often fIsland, B.W.I., in which it was found that lightning is often followed in the ollowed in the cloud by a rapidly intensifying echo and then by a gush of rain cloud by a rapidly intensifying echo and then by a gush of rain at the ground. The at the ground. The increases in radar reflectivity in small volumes of the cloud foincreases in radar reflectivity in small volumes of the cloud following lightning suggest llowing lightning suggest that the electric discharge is influencing the size of particlesthat the electric discharge is influencing the size of particles

in the cloud.in the cloud.An analysis indicates that within 30 seconds after a lightning dAn analysis indicates that within 30 seconds after a lightning discharge, the ischarge, the mass of some droplets may increase as much as 100mass of some droplets may increase as much as 100--fold as the result of an fold as the result of an electrostatic precipitation effect.electrostatic precipitation effect.

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16.2 Electric potential and potential energy 16.2 Electric potential and potential energy due to point chargesdue to point charges

Electric circuits: point of zero potential is defined by Electric circuits: point of zero potential is defined by grounding some point in the circuitgrounding some point in the circuitElectric potential due to a point charge at a point in Electric potential due to a point charge at a point in space: point of zero potential is taken at an infinite space: point of zero potential is taken at an infinite distance from the chargedistance from the chargeWith this choice, a potential can be found asWith this choice, a potential can be found as

Note: the potential depends only on charge of an object, Note: the potential depends only on charge of an object, qq, and a distance from this object to a point in space, , and a distance from this object to a point in space, rr..

eqV kr

=

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Superposition principle for potentialsSuperposition principle for potentials

If more than one point charge is present, their electric If more than one point charge is present, their electric potential can be found by applying potential can be found by applying superposition superposition principleprinciple

The total electric potential at some point P due to several The total electric potential at some point P due to several point charges is the algebraic sum of the electric point charges is the algebraic sum of the electric potentials due to the individual charges.potentials due to the individual charges.

Remember that potentials are scalar quantities!Remember that potentials are scalar quantities!

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Potential energy of a system of point Potential energy of a system of point chargescharges

Consider a system of two particlesConsider a system of two particlesIf VIf V11

is the electric potential due to charge qis the electric potential due to charge q11

at a point P, at a point P, then work required to bring the charge qthen work required to bring the charge q22

from infinity to P from infinity to P without acceleration is qwithout acceleration is q22

VV11

. If a distance between P and . If a distance between P and qq11

is r, then by definitionis r, then by definition

Potential energy is Potential energy is positivepositive

if charges are of the if charges are of the same same signsign

and vice versa.and vice versa.

P A

q1q2

r

1 22 1 e

q qPE q V kr

= =

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MiniMini--quiz: potential energy of an ionquiz: potential energy of an ion

Three ions, Na+, Na+, and Cl-, located such, that they form corners of an equilateral

triangle of side 2 nm in

water. What is the electric potential energy of one of the Na+

ions?

Cl-

Na+ Na+

?[ ]Na Cl Na Na Na

e e e Cl Naq q q q qPE k k k q q

r r r= + = +

but : !Cl Naq q= −

[ ] 0Nae Na Na

qPE k q qr

= − + =

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16.3 Potentials and charged conductors16.3 Potentials and charged conductors

Recall that work is opposite of the change in potential Recall that work is opposite of the change in potential energy,energy,

No work is required to move a charge between two points No work is required to move a charge between two points that are at the same potential. That is, W=0 if Vthat are at the same potential. That is, W=0 if VBB

=V=VA A

Recall: Recall: 1.1.

all charge of the charged conductor is located on its surfaceall charge of the charged conductor is located on its surface2.2.

electric field, E, is always perpendicular to its surface, i.e. electric field, E, is always perpendicular to its surface, i.e. no work is no work is done if charges are moved along the surfacedone if charges are moved along the surface

Thus: potential is constant everywhere on the surface of a Thus: potential is constant everywhere on the surface of a charged conductor in equilibriumcharged conductor in equilibrium

[ ]B AW PE q V V= − = − −

… but that’s not all!

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Because the electric field is zero inside the conductor, no Because the electric field is zero inside the conductor, no work is required to move charges between any two work is required to move charges between any two points, i.e. points, i.e.

If work is zero, any two points inside the conductor have If work is zero, any two points inside the conductor have the same potential, i.e. potential is constant everywhere the same potential, i.e. potential is constant everywhere inside a conductorinside a conductorFinally, since one of the points can be arbitrarily close to Finally, since one of the points can be arbitrarily close to the surface of the conductor, the surface of the conductor, the electric potential is the electric potential is constant everywhere inside a conductor and equal to its constant everywhere inside a conductor and equal to its value at the surface!value at the surface!

Note that the potential inside a conductor is Note that the potential inside a conductor is notnot

necessarily zero, necessarily zero, even though the interior electric field is always zero!even though the interior electric field is always zero!

[ ] 0B AW q V V= − − =

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The electron voltThe electron volt

A unit of energy commonly used in atomic, nuclear and A unit of energy commonly used in atomic, nuclear and particle physics is electron volt (particle physics is electron volt (eVeV))

The electron volt is defined as the energy that electron The electron volt is defined as the energy that electron (or proton) gains when accelerating through a potential (or proton) gains when accelerating through a potential difference of 1 Vdifference of 1 V

Relation to SI:Relation to SI:

1 1 eVeV

= 1.60= 1.60××1010--19 19 CC··V = 1.60V = 1.60××1010--19 19 J J

Vab =1 V

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ProblemProblem--solving strategysolving strategy

Remember that potential is a scalar quantityRemember that potential is a scalar quantitySuperposition principle is an algebraic sum of potentials due Superposition principle is an algebraic sum of potentials due to a system of chargesto a system of chargesSigns are importantSigns are important

Just as in mechanics, only changes in electric potential Just as in mechanics, only changes in electric potential are significant, hence, the point you choose for zero are significant, hence, the point you choose for zero electric potential is arbitrary.electric potential is arbitrary.

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Example : ionization energy of the electron Example : ionization energy of the electron in a hydrogen atomin a hydrogen atom

In the Bohr model of a hydrogen atom, the electron, if it is in the ground state, orbits the proton at a distance of r = 5.29××10-11 m. Find the ionization energy of the atom, i.e. the energy required to remove the electron from the atom.

Note that the Bohr model, the idea of electrons as tiny balls orbiting the nucleus, is not a very good model of the atom. A better picture is one in which the electron is spread out around the nucleus in a cloud of varying density; however, the Bohr model does give the right answer for the ionization energy

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In the Bohr model of a hydrogen atom, the electron, if it is in the ground state, orbits the proton at a distance of r = 5.29 x 10-11

m. Find the ionization energy, i.e. the energy required to remove the electron from the atom.

Given:

r = 5.292 x 10-11 mme = 9.11×10-31

kg mp = 1.67×10-27 kg|e| = 1.60×10-19 C

Find:

E=?

The ionization energy equals to the total energy of the electron-proton system,

E PE KE= +

22 2182.18 10 J -13.6 eV

2 2e e

e ee

m k ee eE k kr m r r

−⎛ ⎞= − + = − = − × ≈⎜ ⎟

⎝ ⎠

The velocity of e can be found by analyzing the force on the electron. This force is the Coulomb force; because the electron travels in a circular orbit, the acceleration will be the centripetal acceleration:

e c cm a F=

2 2

,2e e

e vPE k KE mr

= − =with

or2 2

2 ,e ev em kr r

= or2

2 ,ee

ev km r

=

Thus, total energy is

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16.4 16.4 EquipotentialEquipotential

surfacessurfacesThey are defined as a surface in space on which the potential is the same for every point (surfaces of constant voltage)The electric field at every point of an equipotential

surface is

perpendicular to the surface

convenient to represent by drawing equipotential

lines

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A Positive and Negative Charge( A Dipole)

Two Positive Charges

Examples of Equipotential

Surfaces

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a+Q

b-Q

16.6 The definition of capacitance16.6 The definition of capacitance

Capacitor: two conductors (separated by an insulator)usually oppositely charged

The capacitance, C, of a capacitor is defined as a ratio of the magnitude of a charge on either conductor to the magnitude of the potential difference between the conductors QC

V=

Δ

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1. A capacitor is basically two parallel conducting plates with insulating material in between. The capacitor doesn’t have to look like metal plates.

Capacitor for use in high-performance audio systems.

2. When a capacitor is connected to an external potential, charges flow onto the plates and create a potential difference between the plates.

+ -

- -

3.

Capacitors in circuitssymbolsanalysis follows from

conservation of energy conservation of charge

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Units of capacitanceUnits of capacitance

The unit of C is the farad (F), but most capacitors have values of C ranging from picofarads

to microfarads (pF

to μF).

Recall, micro ⇒10-6, nano

⇒10-9, pico

⇒10-12

If the external potential is disconnected, charges remain on the plates, so capacitors are good for storing charge (and energy).

1 1F C V=

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A

A

+Q

-Q

d

16.7 The parallel16.7 The parallel--plate capacitorplate capacitor

The capacitance of a device The capacitance of a device depends on the geometric depends on the geometric arrangement of the conductorsarrangement of the conductors

where where AA

is the area of one of is the area of one of the plates, the plates, dd

is the separation, is the separation, εε

00

is a constant called the is a constant called the permittivity of free spacepermittivity of free space,,

εε

00

= 8.85= 8.85××1010--12 12 CC22/N/N··mm22

0ACd

ε=

0

14ekπε

=

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A parallel plate capacitor has plates 2.00 m2 in area, separated by a distance of 5.00 mm. A potential difference of 10,000 V is applied across the capacitor. Determine

the capacitancethe charge on each plateThe electric field between the plates

Problem: parallelProblem: parallel--plate capacitorplate capacitor

3 6/ 10000 / 5.0 10 2.0 10 /E V d V m V m−= Δ = × = ×

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A parallel plate capacitor has plates 2.00 m2 in area, separated by a distance of 5.00 mm. A potential difference of 10,000 V is applied across the capacitor. Determine

the capacitancethe charge on each plate

Given:

ΔV=10,000 VA = 2.00 m2

d = 5.00 mm

Find:

C=?Q=?

Solution:

Since we are dealing with the parallel-plate capacitor, the capacitance can be found as

( )2

12 2 20 3

9

2.008.85 105.00 10

3.54 10 3.54

A mC C N md m

F nF

ε −−

−

= = × ⋅×

= × =

( )( )9 53.54 10 10000 3.54 10Q C V F V C− −= Δ = × = ×

Once the capacitance is known, the charge can be found from the definition of a capacitance via charge and potential difference:

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16.8 Combinations of capacitors16.8 Combinations of capacitors

It is very often that more than one capacitor is used in an It is very often that more than one capacitor is used in an electric circuitelectric circuitWe would have to learn how to compute the equivalent We would have to learn how to compute the equivalent capacitance of certain combinations of capacitorscapacitance of certain combinations of capacitors

C1

C2

C3

C5C1

C2

C3

C4

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1 1 1 2 2 2

1 2 1 2 1 2

1 2

1 21 2

1 2

eq

eq

Q C V Q C VQ Q Q Q Q QQC

V V V V V VQ Q Q

C C CV V V

= =+

≡ = = + = +

+ == +

= =

+Q1

−Q1

C1V=Vab

a

b

+Q2

−Q2

C2

a. Parallel combinationa. Parallel combination

1 2V V V= =

Connecting a battery to the parallel combination of capacitors is equivalent to introducing the same potential difference

for both capacitors,

1 2Q Q Q+ =

The total charge

transferred to the system from the battery is the sum of charges

of the two capacitors,

By definition,

Thus, Ceq

would be

1 2eqC C C= + C1C2

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Parallel combination: notesParallel combination: notes

Analogous formula is true for any number of capacitors,Analogous formula is true for any number of capacitors,

It follows that the equivalent capacitance of a parallel It follows that the equivalent capacitance of a parallel combination of capacitors is greater than any of the combination of capacitors is greater than any of the individual capacitorsindividual capacitors

1 2 3 ...eqC C C C= + + + (parallel combination)

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A 3 μF capacitor and a 6 μF capacitor are connected in parallel across an 18 V battery. Determine the equivalent capacitance and total charge deposited.

Problem: parallel combination of capacitorsProblem: parallel combination of capacitors

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A 3 μF capacitor and a 6 μF capacitor are connected in parallel across an 18 V battery. Determine the equivalent capacitance and total charge deposited.

+Q1

−Q1

C1V=Vab

a

b

+Q2

−Q2

C2Given:

V = 18 VC1

= 3 μFC2

= 6 μF

Find:

Ceq

=?Q=?

First determine equivalent capacitance of C1 and C2

:

12 1 2 9C C C Fμ= + =

Next, determine the charge

( )( )6 49 10 18 1.6 10Q C V F V C− −= Δ = × = ×

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b. Series combinationb. Series combination

1

1C

1 2V V V= +

Connecting a battery to the serial combination

of capacitors is equivalent to introducing the same charge

for both capacitors,

1 2Q Q Q= =

A voltage induced in the system from the battery is the sum of potential differences across the individual capacitors,

By definition,

Thus, Ceq

would be

+Q1

−Q1

C1

+Q2

−Q2

C2

V=Vab

a

c

b

1 1 1 2 2 2

1 2 1 2 1 2

1 2

1 2

1 21 2

1

1 1 1eq

eq

Q CV Q CVV V V V V VV

C Q Q Q Q Q QQ Q QV V VC C C

= =+

≡ = = + = +

= == +

+ =1 2

1 1 1

eqC C C= +

1

1C 2

1C

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Series combination: notesSeries combination: notes

Analogous formula is true for any number of capacitors,Analogous formula is true for any number of capacitors,

It follows that the equivalent capacitance of a series It follows that the equivalent capacitance of a series combination of capacitors is always less than any of the combination of capacitors is always less than any of the individual capacitance in the combinationindividual capacitance in the combination

1 2 3

1 1 1 1 ...eqC C C C

= + + + (series combination)

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A 3 μF capacitor and a 6 μF capacitor are connected in series across an 18 V battery. Determine the equivalent capacitance.

Problem: series combination of capacitorsProblem: series combination of capacitors

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A 3 μF capacitor and a 6 μF capacitor are connected in series across an 18 V battery. Determine the equivalent capacitance and total charge deposited.

+Q1

−Q1

C1

+Q2

−Q2

C2

V=Vab

a

c

b

Given:

V = 18 VC1

= 3 μFC2

= 6 μF

Find:

Ceq

=?Q=?

First determine equivalent capacitance of C1 and C2

:

1 2

1 2

2eqC CC F

C Cμ= =

+Next, determine the charge

( )( )6 52 10 18 3.6 10Q C V F V C− −= Δ = × = ×

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12

U QV=

16.9 Energy stored in a charged capacitor16.9 Energy stored in a charged capacitor

Consider a battery connected to a capacitorA battery must do work to move electrons from one plate to the other. The work done

to move a small charge Δq across a voltage V is ΔW = V Δq. As the charge increases, V increases so the work to bring Δq increases. Using calculus we find that the energy (U) stored on a capacitor is given by:

V

V

q

221

2 2Q CVC

= =

Q

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+++++++++++

−−−−−−−−−−−

Find electric field energy density (energy per unit volume) in a parallel-plate capacitor

2

0

20

20

12

/ energy density1= ( ) /( )2

12

volume A

U CV

AC V Edd

u U volumeA Ed d

E

d

Ad

u

ε

ε

ε

=

=

= =

≡ =

=

Example: electric field energy in parallelExample: electric field energy in parallel--plate plate capacitorcapacitor

Recall

Thus,

and so, the energy density is

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C1

C2

V

C3

In the circuit shown V = 48V, C1 = 9μF, C2 = 4μF and C3 = 8μF.(a) determine the equivalent capacitance of the circuit,(b) determine the energy stored in the combination by calculating

the energy stored in the equivalent capacitance.

Example: stored energyExample: stored energy

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C1

C2

V

C3

In the circuit shown V = 48V, C1 = 9μF, C2 = 4μF and C3 = 8μF.(a) determine the equivalent capacitance of the circuit,(b) determine the energy stored in the combination by calculating the energy stored in the equivalent capacitance,

First determine equivalent capacitance of C2 and C3

:

23 2 3 12C C C Fμ= + =

( )( )22 6 31 1 5.14 10 48 5.9 102 2

U CV F V J− −= = × = ×

The energy stored in the capacitor C123 is then

Next, determine equivalent capacitance of the circuit by noting that C1

and C23

are connected in series

1 23123

1 23

5.14eqC CC C F

C Cμ≡ = =

+

Given:

V = 48 VC1

= 9 μFC2

= 4 μFC3

= 8 μF

Find:

Ceq

=?U=?

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+Q +Q−Q −Q

V0 V

16.10 Capacitors with dielectrics16.10 Capacitors with dielectricsA dielectrics is an insulating material (rubber, glass, etc.)Consider an insolated, charged capacitor

Notice that the potential difference decreases (k = V0 /V)Since charge stayed the same (Q=Q0

) → capacitance increases

dielectric constant: k = C/C0

Dielectric constant is a material property

0 0 00

0 0

Q Q QC CV V V

κ κκ

= = = =

Insert a dielectric