Electricity. Electrostatic

The Electric Field Electric charge. Conductors and Insulators Coulomb´s Law The Electric field. Electric Field Lines Calculating Electric Field for continous charge distribution Gauss´s Law.

Electric Potential Potential Difference Calculating Electric Potential for a System of Point Charges and for Continuous Charge distribution Potential vs. Electric Field. Field Lines and Equipotential Surfaces Charge and Field at Conductor Surface. Motion of Point Charges in Electric Field. Electric Dipoles in Electric Field

Electrostatic Energy and Capacitance Electrostatic Potential Energy Capacitance

The Storages of Electric Energy Capacitors, Batteries and Circuits Dielectrics. Molecular View of a Dielectric

The Electric Field

Electric charge

Conductors and Insulators

Coulomb´s Law

The Electric field. Electric Field Lines

Calculating Electric Field for continous charge distribution

Gauss´s Law

Charge and Field at Conductor Surfaces

Motion of Point Charges in Electric Field.

Electric Dipoles in Electric Field

Electric Charge: Objects carrying charges of opposite signs attract

each other; Objects carrying charges of the same sign repel each other

Positive, the charge acquired by a glass rod when is rubbed with a piece of silk, (Franklin criteria), then electrons are transferred to silk. The piece of silk acquires the same Negative charge

Charge QuantizacionLaw of Conservation of Charge

The SI unit of charge is the coulomb [C]

Fundamental Unit of Charge e = 1.602177 x 10-19 C

The girl has been charged by contact with the dome of a Van de Graaff generator. She is isolated from the floor. Could you explain why her hair is as shown in the picture?

Conductors materials where the charges, usually electrons, are free to move about the the entire material, such as copper, iron,.. Insulators materials where the charges can not move freely, such as glass, wood

Charging by Induction

Induction via grounding: using the earth as an infinitely large conductor

An object that has separated equal and opposite charges is said to be polarized What happens with

the charges distribution on the spheres once the rod is removed?

Conductors

Insulators

Coulomb´s Law

Coulomb´s torsion balance

Is the force exerted by q1 on q2

k Coulomb constant: 8.99x109 N. m2/C2

Unit vector pointing from q1 to q2

In a hydrogen atom, the electron is separated from the proton by an average distance of about 5.3x10-11m. Calculate the magnitude of the electrostatic force of attraction. Compare the electrostatic force with the gravity force between the proton and the electron.

Force exerted by a System of Charges: Principle of superposition of forces

Three charges are placed as shown in the figure. Calculate the force exerted on the particle at top of the isosceles triangle. Q= 10 μC; q= 500 nC; d = 10 cm

The Electric Field

To avoid the conceptual problem of the action at a distance –instantaneous transmission- the concept of electric field is introduced

[Suppose that a charged particle at some point is suddenly moved. Does the force exerted on a particle some distance away change instantaneously?]

EqF

q

FE

q is a small positive test charge

The force exerted on the charge q

Derive a general expression for the electric field on a point P due to a single charge Q. P is placed at a distance r of the charge. Estimate the value of the electric field for Q=10 nC and r= 15m.

Electric Field for a system of charges

SI units [N/C] [V/m]

Electric Dipole

A system of two equal and opposite charges separated by a small distance is called a electric dipole .

Its strength and orientation is described by the electric dipole moment

aLLqp 2,

Exercise: Calculate the electric field of dipole in point P in the dipole axis. Consider the situation when x»a.

3

2

x

pkE

The Electric Field Lines, or Lines of Force

At any given point, the field vector E is tangent to the field line. They are also called lines of force because they show the direction of the force exerted on the positive test charge.

The density of the lines (the number of lines per unit of area perpendicular to the lines) at any point is proportional to the magnitude of the field at that point

The Electric Field Lines

The electric field lines for two conducting spheres are shown in the figure. What is the relative sign and magnitudes of the charges on the two spheres?

(A) Picture a uniform vertical electric field E = -2000 N/C. (B)The same but the value of electric field is two times the previous value.

Calculating Electric Field for Continous Charge Distribution

In the macroscopic world charge can usually be described as continuosly distributed.

dl

dQdS

dQV

Q

dV

dQaverage

; Volume charge density

Surface charge density

Linear charge density

When the charge is distributed on a volume

When the charge is distributed on a surface

When the charge is distributed along a line

rr

dqkE

Vˆ

2

Applying Coulomb´s Law and the principle of superposition

Calculating Electric Field for Continous Charge Distribution.

E on the Axis of a finite Line charge

E on the Axis of a finite Line charge

E due to an infinite line charge

Gauss´s LawGauss´s Law is one of the so called Maxwell´s Equations that describe the electromagnetics phenomena. For static charges, Coulomb´s Law and Gauss´s Law are equivalent, but Gauss´s Law is more general.

Gauss´s Law:The net number of lines out of any surface enclosing the charges is proportional to the net charge enclosed by the surface

Gauss´s Law can be used to calculate the electric field for charge distribution with high degree of symmetry

Gauss´s Law

Electric Flux ϕ The number of field lines penetrating a surface is called the electric flux. Units: N. m2/C

ndSEdSEd

AnE

AE

If we consider a surface A perpendicular to E,

In the case of surface that is not perpendicular to E, the dot product enables us to obtain the value of area perpendicular to the electric field.

Calculating E from Gauss´s Law. The power of symmetry

Electric field for a single point chargeThe electric field exhibits spherical symmetry around the charge. Then we consider a spherical surface with center on the charge to apply Gauss´s law. The value of E is constant in all points of this sphere.

2

2 44

R

QkE

QkREdAEA

o

insidenet

o

o

Q

rr

qE

k

ˆ4

1

4

1

2

The flux is independent from the selected sphere

Writing Gauss´s Law and Coulomb´s Law in terms of permitivity of free space

2

2121085.84

1mN

Ck

xo

Calculating E from Gauss´s Law. The power of symmetry

Electric field for a Thin Spherical Shell of Charge

The electric field exhibits spherical symmetry around the uniform charge distribution . Then we consider a spherical surface with the same center as the shell of charge to apply Gauss´s law. The value of E is constant in all points of this sphere.

0

0

E

dAEA

The flux is independent from the selected sphere

22

2

4

1

44

R

Q

R

QkE

QkREdAE

o

A

For a gaussian sphere inside of the shell charge

In the Earth´s atmosphere, the electric field is 150 N/C downwards at an altitude of 250 m, and 170 N/C downwards at an altitude of 400 m. Calculate the volume charge density of the atmosphere assuming it to be uniform between both altitudes.

Electric Potential Potential Difference Calculating Electric Potential for a System of Point Charges and for Continuous Charge distribution Potential vs. Electric Field. Field Lines and Equipotential Surfaces Charge and Field at Conductor Surface. Motion of Point Charges in Electric Field. Electric Dipoles in Electric Field

Potential differenceElectrostatic force is a conservative force, therefore the change in the potential energy, U, is given by

21

2

,1

2

,1

2

,1UUdlEqdlEqdlF

linelineline

The potential energy per unit of charge, positive, called the potential difference (voltage) is:

q

UV

dlEdVdlEV

VVVq

UU

line

2

,1

2121 Units:

Volt (V)= 1 J/C

1 N/C = 1 V/m

1 eV= 1.60x10-19 J

electron volt [eV]

dlFdU

For convenience, the electric potential and the potential energy of a test charge are chosen to be zero in the same point

Potential energy in the gravitational field

Potential energy in the electric field

Calculating potential difference (voltage)

Uniforme Electric Field Single Point Charge q

Draw the uniform electric field between the two plates shown in the figure E = 1200 N/mTo calculate the potential difference between A and B points. To calculate the potential energy adquired by a positive charge of 1 µC to carry out it from A to B

3 m

A

B

A

B

q

+++++++++++++++++++++

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Q = 25 µC; rA= 1 m; rB= 3 m

Potential Due to a System of Point Charges

r

kqV

r

kq

r

kqV

ref

Chosing the reference point infinitely far from the point charge rref ≈∞

The potential energy U of a test charge qo placed at

distance r from the point charge q is

r

qqkVqU o

o

Potential from a single point charge

ELECTROSTATIC POTENTIAL ENERGY OF A

TWO-CHARGE SYSTEM

Computing the Electric Field from the Potential

EVVgrad

ldEdV

Calculations of V for Continuous Charge Distributions

r

dqkV

Potential for a Thin Spherical Shell of Charge

Equipotetntial surfaces