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Chương 1 Từ trường PHYSICS 1 MECHANICSAND THERMODYNAMICS PHYSICS 2 ELECTRICITY, MAGNETISM, OPTICS, AND MODERN PHYSICS PART 4 Electricity and Magnetism Chapter 1 Electric Fields Chapter 2 Gauss’s Law Chapter 3 Electric Potential Chapter 4 Capacitance and Dielectrics Chapter 5 Current and Resistance Chapter 6 Direct Current Circuits Chapter 7 Magnetic Fields Chapter 8 Sources of the Magnetic Field Chapter 9 Faraday’s Law CHAPTER 1 (3) ELECTRIC FIELDS 1 1 Properties of Electric Charges 1 2 Charging.

PHYSICS 1: MECHANICS AND THERMODYNAMICS

PHYSICS 2: ELECTRICITY, MAGNETISM, OPTICS,

AND MODERN PHYSICS

Electricity and Magnetism

Chapter 1: Electric Fields

Chapter 2: Gauss’s Law

Chapter 3: Electric Potential

Chapter 4: Capacitance and Dielectrics

Chapter 5: Current and Resistance

Chapter 6: Direct-Current Circuits

Chapter 7: Magnetic Fields

Chapter 8: Sources of the Magnetic Field

Chapter 9: Faraday’s Law

CHAPTER 1 (3)

ELECTRIC FIELDS

1.1 Properties of Electric Charges

1.2 Charging Objects by Induction

1.7 Motion of a Charged Particle

in a Uniform Electric Field

1.1 Properties of Electric Charges

Charge interaction:

Charge of the same

sign repel one

 Positive ion: 𝑞+ = 𝑁𝑒, Negative ion: 𝑞− = −𝑁𝑒

1.1 Properties of Electric ChargesCHAPTER 1 - ELECTRIC FIELDS

Electric charge always occurs as integral multiples of afundamental amount of charge 𝑒 (quantized):

𝑞 = ±𝑁𝑒

 Neutron: 𝑞𝑛 = 0, Proton: 𝑞𝑝 = 𝑒, Electron: 𝑞𝑒 = −𝑒

1.1 Properties of Electric Charges

Three objects are brought close to each other, two at a time.When objects A and B are brought together, they repel Whenobjects B and C are brought together, they also repel Which ofthe following are true?

(a) Objects A and C possess charges of the same sign

(b) Objects A and C possess charges of opposite sign

(c) All three objects possess charges of the same sign

(d) One object is neutral

(e) Additional experiments must be performed to determine thesigns of the charges

1.2 Charging Objects by InductionCHAPTER 1 - ELECTRIC FIELDS

Electrical conductors are materials in which some of theelectrons are free electrons that are not bound to atoms andcan move relatively freely through the material Ex.: copper,aluminum, silver,…

Electrical insulators are materials in which all electrons arebound to atoms and cannot move freely through thematerial Ex.: glass, rubber, dry wood,…

Semiconductors are a third class of materials, and theirelectrical properties are somewhere between those ofinsulators and those of conductors Ex.: Silicon,germanium,…

1.2 Charging Objects by Induction

Three objects are brought close to one another, two at a time.When objects A and B are brought together, they attract Whenobjects B and C are brought together, they repel Which of thefollowing are necessarily true?

(a) Objects A and C possess charges of the same sign

(b) Objects A and C possess charges of opposite sign

(c) All three objects possess charges of the same sign

(d) One object is neutral

(e) Additional experiments must be performed to determineinformation about the charges on the objects

1.2 Charging Objects by Induction

Electric force between two stationary point charges

(called electrostatic force or Coulomb force):

𝜀0 = 8.854 × 10−12 C2/m2N

 𝑞1 , 𝑞2 : magnitude of point charges

 𝑟: distance between two charges

 Point charge: charged particle of zero size

1.3 Coulomb’s Law

Example 1.1

The electron and proton of a hydrogen atom are separated (on the average)

by a distance of approximately 5.3 × 10 -11 m Find the magnitudes of the electric force and the gravitational force between the two particles.

CHAPTER 1 - ELECTRIC FIELDS

1.3 Coulomb’s Law

Vector form of Coulomb’s law:

The electric force exerted by a charge

When more than two charges are

present, for example, if four charges

are present, the resultant force

exerted by particles 2, 3, and 4 on

particle 1 is

𝑭𝟏 = 𝑭𝟐𝟏 + 𝑭𝟑𝟏 + 𝑭𝟒𝟏

1.3 Coulomb’s Law

1 3

The Superposition Principle

CHAPTER 1 - ELECTRIC FIELDS

1.3 Coulomb’s Law

1.3 Coulomb’s Law

Object A has a charge of +2 µC, and object B has a charge of +6 µC Which statement is true about the electric forces on the objects?

Example 1.2

Consider three point

charges located at the

corners of a right triangle as

shown in the below figure,

1.3 Coulomb’s Law

Example 1.2

1 7

Find the magnitude and

direction of the resultant

electric force on q0

CHAPTER 1 - ELECTRIC FIELDS

1.3 Coulomb’s Law

EX2: Two identical small

charged spheres, each having

a mass of 3×10-2 kg, hang in

equilibrium as shown in

Figure The length L of each

string is 0.150 m, and the

angle  is 50 Find the

magnitude of the charge on

each sphere

 Electric field vector 𝑬

The electric force on the test charge per

unit charge at a point in space is defined

as the electric force acting on a positive

test charge placed at that point divided

by the test charge:

test charge

 If an arbitrary charge 𝑞 is placed in an electric field 𝐸, it

experiences an electric force given by:

𝐹𝑒 = 𝑞𝐸

( 𝐹𝑒: electric force exerts on a test charge 𝑞0)

CHAPTER 1 - ELECTRIC FIELDS

1.4 Analysis Model: Particle in a Field

 Electric field due to a point charge:

The electric field due to a point charge 𝑞 at the location P having a

distance 𝑟 from the charge is

𝑬 = 𝒌𝒆 𝒒

𝒓𝟐 𝒓

𝑟: unit vector direct from 𝑞

2 1

CHAPTER 1 - ELECTRIC FIELDS

1.4 Analysis Model: Particle in a Field

2 2

EX3: A point charge q1

= 8 nC is at the origin

and a second point

charge q2 = 12 nC is on

the axis at x= 4 m Find

the electric field on the

y axis at y = 3 m.

1.4 Analysis Model: Particle in a Field

2 3

EX4: A charge +q is at

x = a and a second

charge –q is at x = -a.

(a) Find the electric

field on the axis at an

arbitrary point x > a.

(b) Find the limiting

form of the electric

field for x >> a.

CHAPTER 1 - ELECTRIC FIELDS

1.4 Analysis Model: Particle in a Field

2 4

Action of the Electric Field on charges

1 Electron moving parallel to a uniform electric field

in the direction of the field.

How far does the electron

travel before it is brought

momentarily to rest?

1.4 Analysis Model: Particle in a Field

2 5

2 Electron moving perpendicular to a uniform electric field

EX6: An electron enters a uniform

electric field E = 2000 (N/C) with

an initial velocity v0 = 1106 (m/s)

perpendicular to the field

(a) Compare the gravitational

force acting on the electron to the

electric force acting on it

(b) By how much has the electron

been deflected after it has traveled

1.0 cm in the x direction?

CHAPTER 1 - ELECTRIC FIELDS

1.4 Analysis Model: Particle in a Field

Action of the Electric Field on charges

 Electric field due to a continuous

2 7

1 Continuous Sources: Charge density

CHAPTER 1 - ELECTRIC FIELDS

1.5 Electric Field of a Continuous Charge

Distribution

2 8

2 Electric field due to a line charge of finite length

EX7: A charge Q is uniformly distributed along the z axis,

from z = -L/2 to z = L/2 Show that for large value of z the expression for the electric field of the line charge on the z axis approaches the expression for the electric field of

a point charge Q at the origin

1.5 Electric Field of a Continuous Charge

Distribution

2 9

EX8: A charge Q is uniformly distributed along the z axis,

from z=-L/2 to z=L/2.

(a) Find an expression for the electric field on the z=0

plane as a function of R, the radial distance of the field

point from the axis.

(b) Show that for R>>L, the expression found in Part (a)

approaches that of a point charge at the origin of charge

Q.

(c) Show that for the expression found in Part (a)

approaches that of an infinitely long line charge on the

axis with a uniform linear charge density =Q/L.

CHAPTER 1 - ELECTRIC FIELDS

1.5 Electric Field of a Continuous Charge

Distribution

2 Electric field due to a line charge of finite length

3 0

1.5 Electric Field of a Continuous Charge

Distribution

2 Electric field due to a line charge of finite length

3 1

3 Electric field on the axis of a charged ring.

EX9: A thin ring (a circle)

of radius a is uniformly

charged with total charge

Q Find the electric field

due to this charge at all

points on the axis

perpendicular to the

plane and through the

center of the ring.

CHAPTER 1 - ELECTRIC FIELDS

1.5 Electric Field of a Continuous Charge

Distribution

3 2

4 Electric field on the axis of a charged Disk.

EX10: Consider a uniformly charged thin disk of radius b

and surface charge density  ,

(a) Find the electric field at all points on the axis of the disk.

(b) Show that for points on the axis and far from the disk, the electric field approaches that of a point charge at the origin with the same charge as the disk.

(c) Show that for a uniformly charged disk of infinite radius, the electric field is uniform throughout the region

on either side of the disk.

1.5 Electric Field of a Continuous Charge

Distribution

 When a particle of charge q and mass m is placed in an electric field 𝑬, the electric force exerted on the charge is q𝑬 If that is

the only force exerted on the particle, it must be the net force,and it causes the particle to accelerate Therefore,

 If q >0, its acceleration is in the direction of the electric field

 If q <0, its acceleration is in the direction opposite the electricfield

CHAPTER 1 - ELECTRIC FIELDS

1.7 Motion of a Charged Particle in a

Uniform Electric Field

PHYSICS 2: ELECTRICITY, MAGNETISM, OPTICS,

AND MODERN PHYSICS

CHAPTER 2 (2)

GAUSS’S LAW

2.1 Electric Field Lines and

Electric Flux 2.2 Gauss’s Law 2.3 Application of Gauss’s

Law to Various Charge Distributions

2.4 Conductors in

Electrostatic Equilibrium

 Electric field vector 𝑬 and Electric field lines

Electric field lines (used to visualize

electric field patterns) are related to

the electric field:

 The electric field vector 𝐸 is tangent

to the electric field line at each point.

 The direction of the electric field line

is the same as of 𝐸

 The number of lines per unit area

through a surface perpendicular to

the lines is proportional to the

magnitude of the electric field in that

Electric Field lines are not Paths of Particles.

2.1 Electric Field Lines and Electric Flux

The rules for drawing electric field lines:

 The lines must begin on a positive

charge and terminate on a negative

charge In the case of an excess of one

type of charge, some lines will begin

or end infinitely far away

 The number of lines drawn leaving a

positive charge or approaching a

negative charge is proportional to the

magnitude of the charge

 No two field lines can cross

2.1 Electric Field Lines and Electric FluxCHAPTER 2: GAUSS’S LAW

 Electric field vector 𝑬 and Electric field lines

2.1 Electric Field Lines and Electric Flux

 Electric field vector 𝑬 and Electric field lines

2.1 Electric Field Lines and Electric FluxCHAPTER 2: GAUSS’S LAW

 Electric field vector 𝑬 and Electric field lines

(a) The electric field lines for two point charges of equal magnitude and opposite sign (an electric dipole) The number of lines leaving the positive charge equals the number terminating at the negative charge (b) The dark lines are small pieces of thread suspended in oil, which align with the electric field of a dipole.

2.1 Electric Field Lines and Electric Flux

 Electric field vector 𝑬 and Electric field lines

(a) The electric field lines for two positive point charges (b) Pieces of thread suspended in oil, which align with the electric field created by two equal- magnitude positive charges.

2.1 Electric Field Lines and Electric FluxCHAPTER 2: GAUSS’S LAW

 Electric field vector 𝑬 and Electric field lines

The electric field lines for

a point charge +2q and asecond point charge -q

2.1 Electric Field Lines and Electric Flux

 Electric field vector 𝑬 and Electric field lines

Rank the magnitudes of the electric field at points

A, B, and C (greatest magnitude first).

 Electric flux: 𝚽𝐄 = 𝑬 𝑨

where E is the magnitude of

electric field

A is the surface area

perpendicular to the field

 𝚽𝐄 is proportional to the

number of electric field

lines that penetrate

surface.

2.1 Electric Field Lines and Electric FluxCHAPTER 2: GAUSS’S LAW

A

𝑬

 If electric field is uniform, electric

flux of 𝐸 through an area 𝐴:

𝚽𝑬 = 𝑬𝑨⊥ = 𝑬 𝒏𝑨 = 𝑬𝑨 𝐜𝐨𝐬 𝜽

where 𝐴⊥ is a projection of area 𝐴

perpendicular to the field, and 𝜃 =

 Electric flux 𝚽𝐄

2.1 Electric Field Lines and Electric FluxCHAPTER 2: GAUSS’S LAW

Note:

1 The dependence of electric

flux on the direction of 𝒏:

• 𝜽 < 𝟗𝟎°: 𝚽𝑬 > 𝟎

• 𝜽 > 𝟗𝟎°: 𝚽𝑬 < 𝟎

• 𝜽 = 𝟗𝟎°: 𝚽𝑬 = 𝟎

2 Convention of direction of the

area vector in the case of a

closed area: point outward from

the surface

2.1 Electric Field Lines and Electric Flux

Suppose a point charge is located at the center of a spherical surface The electric field at the surface of the sphere and the total flux through the sphere are determined Now the radius of the sphere is halved What happens to the flux through the sphere and the magnitude of the electric field at the surface of the sphere?

(a) The flux and field both increase.

(b) The flux and field both decrease.

(c) The flux increases, and the field decreases.

(d) The flux decreases, and the field increases.

(e) The flux remains the same, and the field increases.

(f) The flux decreases, and the field remains the same.

2.1 Electric Field Lines and Electric Flux

Example 2.1

2.2 Gauss’s Law CHAPTER 2: GAUSS’S LAW

is independent ofthe shape of thatsurface.

2.2 Gauss’s Law

Determine the net flux through the surfaces S, S ’ , and S ’’

2.2 Gauss’s Law CHAPTER 2: GAUSS’S LAW

2.2 Gauss’s Law

2.2 Gauss’s Law CHAPTER 2: GAUSS’S LAW

If the net flux through a gaussian surface is zero, the following four statements could be true Which of the statements must be true?

(a) There are no charges inside the surface.

(b) The net charge inside the surface is zero.

(c) The electric field is zero everywhere on the surface.

(d) The number of electric field lines entering the surface equals the number leaving the surface.

Gauss’s law is useful for determining electric fields when the charge distribution is highly symmetric so that we can choose a gaussian surface satisfying one or more of the following conditions:

1 The value of the electric field can be argued by symmetry to be

constant over the portion of the surface.

2 𝐸 and 𝑑 𝐴 are parallel → Φ𝐸 = 𝑆 𝐸 𝑑𝐴

3 𝐸 and 𝑑 𝐴 are perpendicular over a portion of the surface.

4 The electric field is zero over the portion of the surface.

2.3 Application of Gauss’s Law to Various Charge

Distributions

Note: Gaussian Surfaces Are not Real

A gaussian surface is an imaginary surface you construct to satisfy the conditions listed here It does not have to coincide with a physical surface in the situation.

2 2

2.3 Application of Gauss’s Law to Various Charge

Distributions

Example 2.2

(A) Calculate the magnitude of the electric field

at a point outside the sphere.

2.3 Application of Gauss’s Law to Various Charge

Distributions

CHAPTER 2: GAUSS’S LAW

Example 2.2

(B) Find the magnitude of the electric

field at a point inside the sphere.

2.3 Application of Gauss’s Law to Various Charge

Distributions

Example 2.3

 Because the charge is distributed

uniformly along the line, the charge

symmetry and we can apply Gauss’s

law to find the electric field.

distribution requires that 𝐸 be

perpendicular to the line charge and

directed outward as shown in the

figure.

2.3 Application of Gauss’s Law to Various Charge

Distributions

 To reflect the symmetry of the charge distribution, let’s choose a cylindrical

gaussian surface of radius r and length ℓ, that is coaxial with the line

charge For the curved part of this surface, 𝐸 is constant in magnitude and perpendicular to the surface at each point, satisfying conditions (1) and (2) Furthermore, the flux through the ends of the gaussian cylinder is zero because 𝐸 is parallel to these surfaces That is the first application we have seen of condition (3).