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General Physics (PHY 2140)

Lecture 4Lecture 4Electrostatics

Electric flux and Gauss’s lawElectrical energy

potential difference and electric potential

potential energy of charged conductors

Chapters 15-16

http://www.physics.wayne.edu/~apetrov/PHY2140/

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Lightning ReviewLightning Review

Last lecture:

1. Properties of the electric field, field lines2. Conductors in electrostatic equilibrium

Electric field is zero everywhere within the conductor.Electric field is zero everywhere within the conductor.Any excess charge field on an isolated conductor resides on its Any excess charge field on an isolated conductor resides on its surface.surface.The electric field just outside a charged conductor is perpendicThe electric field just outside a charged conductor is perpendicular to the ular to the conductor’s surface.conductor’s surface.On an irregular shaped conductor, the charge tends to accumulateOn an irregular shaped conductor, the charge tends to accumulate at at locations where the radius of curvature of the surface is smallelocations where the radius of curvature of the surface is smallest.st.

Review Problem: Would life be different if the electron were positively charged and the proton were negatively charged? Does the choice of signs have any bearing on physical and chemical interactions?

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15.10 Electric Flux and Gauss’s Law15.10 Electric Flux and Gauss’s Law

A convenient technique was introduced by Karl F. Gauss A convenient technique was introduced by Karl F. Gauss (1777(1777--1855) to calculate electric fields.1855) to calculate electric fields.Requires symmetric charge distributions.Requires symmetric charge distributions.Technique based on the notion of Technique based on the notion of electrical fluxelectrical flux..

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15.10 Electric Flux15.10 Electric Flux

To introduce the notion of flux, consider To introduce the notion of flux, consider a situation where the electric field is a situation where the electric field is uniform in magnitude and direction. uniform in magnitude and direction. Consider also that the field lines cross a Consider also that the field lines cross a surface of area A which is surface of area A which is perpendicular to the field.perpendicular to the field.The number of field lines per unit of The number of field lines per unit of area is constant.area is constant.The flux, The flux, ΦΦ, is defined as the product of , is defined as the product of the field magnitude by the area crossed the field magnitude by the area crossed by the field lines.

Area=A

Eby the field lines.

EAΦ =

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15.10 Electric Flux15.10 Electric Flux

Units: NmUnits: Nm22/C in SI units./C in SI units.

Find the electric flux through the area A = 2 mFind the electric flux through the area A = 2 m22, which is , which is perpendicular to an electric field E=22 N/Cperpendicular to an electric field E=22 N/C

EAΦ =Answer: Answer: ΦΦ = 44 Nm2/C.= 44 Nm2/C.

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15.10 Electric Flux15.10 Electric Flux

If the surface is not perpendicular to the field, the If the surface is not perpendicular to the field, the expression of the field becomes:expression of the field becomes:

Where Where θθ is the angle between the field and a normal to is the angle between the field and a normal to the surface.the surface.

cosEA θΦ =

θθ

N

θθ

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15.10 Electric Flux15.10 Electric Flux

Remark: Remark: When an area is constructed such that a closed surface When an area is constructed such that a closed surface is formed, we shall adopt the convention that the flux is formed, we shall adopt the convention that the flux lines passing lines passing intointo the interior of the volume are the interior of the volume are negativenegativeand those passing and those passing out ofout of the interior of the volume are the interior of the volume are positivepositive..

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Example:Example:Question:Question:Calculate the flux of a constant E field (along x) through a Calculate the flux of a constant E field (along x) through a cube of side “L”.cube of side “L”.

x

y

z

E1 2

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Question:Question:Calculate the flux of a constant E field (along x) through a cubCalculate the flux of a constant E field (along x) through a cube of side “L”.e of side “L”.Reasoning:Reasoning:

Dealing with a composite, closed surface.Dealing with a composite, closed surface.Sum of the fluxes through all surfaces.Sum of the fluxes through all surfaces.Flux of field going in is negativeFlux of field going in is negativeFlux of field going out is positive.Flux of field going out is positive.E is parallel to all surfaces except surfaces labeled 1 and 2.E is parallel to all surfaces except surfaces labeled 1 and 2.So only those surface contribute to the flux.So only those surface contribute to the flux.

Solution:Solution:

x

y

z

E1 2

2 2 0net EL ELΦ = − + =

21 1 1cosEA ELθΦ = − = −

22 2 2cosEA ELθΦ = =

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15.10 Gauss’s Law15.10 Gauss’s Law

The net flux passing through a closed surface The net flux passing through a closed surface surrounding a charge Q is proportional to the magnitude surrounding a charge Q is proportional to the magnitude of Q:of Q:

In free space, the constant of proportionality is 1/In free space, the constant of proportionality is 1/εεoowhere where εεoo is called the permittivity of of free space.

cosnet EA QθΦ = ∝∑

is called the permittivity of of free space.

( )12 2 2

9 2 2

1 1 8.85 104 4 8.99 10 /o

e

C N mk Nm C

επ π

−= = = × ⋅×

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15.10 Gauss’s Law15.10 Gauss’s Law

The net flux passing through any closed surface is equal The net flux passing through any closed surface is equal to the net charge inside the surface divided by to the net charge inside the surface divided by εεoo..

Can be used to compute electric fields. Example: point Can be used to compute electric fields. Example: point chargecharge

cosneto

QEA θε

Φ = =∑

22 2

0

cos 44net e

Q QEA r E E kr r

θ ππε

Φ = = → = =∑

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16.0 Introduction16.0 Introduction

The Coulomb force is a conservative forceThe Coulomb force is a conservative forceA potential energy function can be defined for any A potential energy function can be defined for any conservative force, including Coulomb forceconservative force, including Coulomb forceThe notions of potential and potential energy are The notions of potential and potential energy are important for practical problem solvingimportant for practical problem solving

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16.1 Potential difference and electric potential16.1 Potential difference and electric potential

The electrostatic force is The electrostatic force is conservativeconservativeAs in mechanics, work isAs in mechanics, work is

Work done on the positive charge Work done on the positive charge by moving it from A to B

E

cosW Fd ϑ=BA

d by moving it from A to B

cosW Fd qEdϑ= =

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Potential energy of electrostatic fieldPotential energy of electrostatic field

The work done by a conservative force equals the The work done by a conservative force equals the negativenegative of the change in potential energy, of the change in potential energy, ∆∆PEPE

This equation This equation is valid only for the case of a is valid only for the case of a uniformuniform electric fieldelectric fieldallows to introduce the concept of allows to introduce the concept of electric potential

PE W qEd∆ = − = −

electric potential

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Electric potentialElectric potential

The The potential differencepotential difference between points A and B, Vbetween points A and B, VBB--VVAA, , is defined as the change in potential energy (final minus is defined as the change in potential energy (final minus initial value) of a charge, q, moved from A to B, divided initial value) of a charge, q, moved from A to B, divided by the chargeby the charge

Electric potential is a Electric potential is a scalar quantityscalar quantityElectric potential difference is a measure of electric Electric potential difference is a measure of electric energy per unit chargeenergy per unit chargePotential is often referred to as “voltage”

B APEV V Vq

∆∆ = − =

Potential is often referred to as “voltage”

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Electric potential Electric potential -- unitsunits

Electric potential difference is the work done to move a Electric potential difference is the work done to move a charge from a point A to a point B divided by the charge from a point A to a point B divided by the magnitude of the charge. Thus the SI units of electric magnitude of the charge. Thus the SI units of electric potentialpotential

In other words, 1 J of work is required to move a 1 C of In other words, 1 J of work is required to move a 1 C of charge between two points that are at potential charge between two points that are at potential difference of 1 V

1 1V J C=

difference of 1 V

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Electric potential Electric potential -- notesnotes

Units of electric field (N/C) can be expressed in terms of the Units of electric field (N/C) can be expressed in terms of the units of potential (as volts per meter)units of potential (as volts per meter)

Because the positive tends to move in the direction of the Because the positive tends to move in the direction of the electric field, work must be done on the charge to move it in electric field, work must be done on the charge to move it in the direction, opposite the field. Thus,the direction, opposite the field. Thus,

A positive charge gains electric potential energy when it is movA positive charge gains electric potential energy when it is moved in ed in a direction opposite the electric fielda direction opposite the electric fieldA negative charge looses electrical potential energy when it movA negative charge looses electrical potential energy when it moves es in the direction opposite the electric field

1 1N C V m=

in the direction opposite the electric field

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Analogy between electric and gravitational fieldsAnalogy between electric and gravitational fields

The same kineticThe same kinetic--potential energy theorem works herepotential energy theorem works here

If a positive charge is released from A, it accelerates in If a positive charge is released from A, it accelerates in the direction of electric field, i.e. gains kinetic energythe direction of electric field, i.e. gains kinetic energyIf a negative charge is released from A, it accelerates in If a negative charge is released from A, it accelerates in the direction opposite the electric fieldthe direction opposite the electric field

A

B

qd

A

B

mdE g

i i f fKE PE KE PE+ = +

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Example: motion of an electronExample: motion of an electron

What is the speed of an electron accelerated from rest across a potential difference of 100V? What is the speed of a proton accelerated under the same conditions?

Observations: 1. given potential energy

difference, one can find the kinetic energy difference

2. kinetic energy is related to speed

Given:

∆V=100 Vme = 9.11×10-31 kg mp = 1.67×10-27 kg|e| = 1.60×10-19 C

Find:ve=?vp=?

Vab

i i f fKE PE KE PE+ = +

f i fKE KE KE PE q V− = = ∆ = ∆

21 22 f f

q Vmv q V vm∆

= ∆ → =

6 55.9 10 , 1.3 10e pm mv vs s= × = ×

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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 q22VV11. 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

1 22 1 e

q qPE q V kr

= =q2 q1

r

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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 Cl Na Na Nae e e Cl Na

q q q q qPE k k k q qr r r

= + = +?

but : !Cl Naq q= −

[ ] 0Nae Na Na

qPE k q qr

= − + =Na+ Na+

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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 in zero inside the conductor, Because the electric field in zero inside the conductor, no work is required to move charges between any two no 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 surfacevalue 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!

[ ] 0B AW q V V= − − =

even though the interior electric field is always zero!

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

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

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=?

2 2

,2e

e vPE k KE mr

= − =E PE KE= + with

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:

c cma F= or2 2

2 ,ev em kr r= or

22 ,e

ev kmr

=

Thus, total energy is

22 2182.18 10 J -13.6 eV

2 2e

e ek ee m eE k k

r mr r−

= − + = − = − × ≈

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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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