Electricity and Magnetism - personales.unican.espersonales.unican.es/lopezqm/FBE/elmenu/teoria/FBE=Tema VIII_20121… · Fundamental Forces of Nature The force of gravity has been

Electricity and MagnetismElectricity and MagnetismElectricity and MagnetismElectricity and MagnetismElectricity and MagnetismElectricity and MagnetismElectricity and MagnetismElectricity and Magnetism

� Electricity and magnetism have been known for thousands of years• The ancient Greeks knew that a piece of amber rubbed with fur would attract small, light objects• The word for electron and electricity derived from the Greek word for amber, ηλεκτρον

• Naturally occurring magnetic materials called lodestones

10-ene-13 Tema VIII. Electrostática (1) 1

• Naturally occurring magnetic materials called lodestones were used as early as 300 BC to construct compasses

� The relationship between electricity and magnetism was not known until the middle of the 19th century

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Fundamental Forces of NatureFundamental Forces of NatureFundamental Forces of NatureFundamental Forces of NatureFundamental Forces of NatureFundamental Forces of NatureFundamental Forces of NatureFundamental Forces of Nature

� The force of gravity has been known since the time of Newton• Late 17th century

� In the 20th century, two more forces were discovered• The weak force and the strong force

� Around 1970 the electromagnetic force and the weak force were unified• The electroweak force


• The electroweak force

• 1979 Nobel prize in physics for Weinberg, Salam, and Glashow

� Currently physicists are workingto unify the electroweak forceand the strong force

� Gravity remains a puzzle although it was identified first

The Four ForcesThe Four ForcesThe Four ForcesThe Four ForcesThe Four ForcesThe Four ForcesThe Four ForcesThe Four Forces

� In our model of the world, the four fundamental forces work by exchanging elementary particles• Gravity - graviton(has not been observed yet)

• Electromagnetic - photon• Weak -W and Z bosons(observed in 1983)


Weak -W and Z bosons(observed in 1983)

• Strong – gluons(observed in 1979)

� Thus forces can act a distance without touching• The Sun can attract the Earth from 93 million miles away

• Magnet can attract metal

The Fundamental ForcesThe Fundamental ForcesThe Fundamental ForcesThe Fundamental ForcesThe Fundamental ForcesThe Fundamental ForcesThe Fundamental ForcesThe Fundamental ForcesRange:

Infinite

Infinite


10-18 m0.1% of the diameter of a proton

10-15 mdiameter of medium-sized nucleus

Gravitational and Electric ForcesGravitational and Electric ForcesGravitational and Electric ForcesGravitational and Electric ForcesGravitational and Electric ForcesGravitational and Electric ForcesGravitational and Electric ForcesGravitational and Electric Forces

� For gravity we defined a gravitational force

� And a gravitational potential

F(r) = Gm1m2

r2


� We will do the same for the electric force and the electric potential

� We will introduce the concept of an electric field to help us understand the electromagnetic force

U(r) = −Gm1m2

r

Electric ChargeElectric ChargeElectric ChargeElectric ChargeElectric ChargeElectric ChargeElectric ChargeElectric Charge

� Everyday example: When walking on a carpet on a dry winter’s day and then touching a door knob, one often experiences a spark• This process is called charging

• Charging: negatively charged electrons move from the atoms and molecules of the carpet to the soles of our shoes, to the body

• Spark: The built-up charge discharges through the metal of the door knob


knob

� Similar phenomenon involving wind, rain and ice produces lightning

Charge (2)Charge (2)Charge (2)Charge (2)Charge (2)Charge (2)Charge (2)Charge (2)

� Normally objects around us do not seem to carry a net charge� They have equal amounts of positive and negative charge and are thus

electrically neutral

� Demo:

� Negative charge: an excess of electrons� Positive charge: a deficit of electrons


� Demo:• If we rub a plastic rod with fur, the rod will become charged

• If we bring two charged plastic rods together, they will repel each other

• If we rub a glass rod with silk, the rod will become charged• If we bring together a charged plastic rod and a charged glass rod, they will attract each other

Measuring Charge: The ElectroscopeMeasuring Charge: The ElectroscopeMeasuring Charge: The ElectroscopeMeasuring Charge: The ElectroscopeMeasuring Charge: The ElectroscopeMeasuring Charge: The ElectroscopeMeasuring Charge: The ElectroscopeMeasuring Charge: The Electroscope

Metal

Metal deflection arm

Metal rod

Electric repulsion

Shield


The glass and the plastic rod have opposite charge

Metal rod

Gravity

Explanation of the DemosExplanation of the DemosExplanation of the DemosExplanation of the DemosExplanation of the DemosExplanation of the DemosExplanation of the DemosExplanation of the Demos

� Explanation:

Electrons are transferred from the fur onto the plastic rod. This rod now carries a negative charge.

Electrons are transferred from the glass rod onto the silk. The glass rod now carries a positive charge (electrons are


The glass rod now carries a positive charge (electrons are missing).

The electroscope shows the presence of charge.

This result leads to the Law of Charges• Like charges repel and opposite charges attract

Law of ChargesLaw of ChargesLaw of ChargesLaw of ChargesLaw of ChargesLaw of ChargesLaw of ChargesLaw of Charges

+

-

+

-

-

+


� Note that electricity is different from gravitation, in which the force is always attractive

+ -

m2m1

Static ClingStatic ClingStatic ClingStatic ClingStatic ClingStatic ClingStatic ClingStatic Cling

� What is the force between an electrically charged object (q) and a neutral object (0)?

� Observe: It is always attractive

Why?


� Why?+q

0

+++++

-----Polarization

The Unit of ChargeThe Unit of ChargeThe Unit of ChargeThe Unit of ChargeThe Unit of ChargeThe Unit of ChargeThe Unit of ChargeThe Unit of Charge

� The unit of charge is the coulomb, abbreviated C[named after Charles-Augustin de Coulomb (1736 -1806)].

� The coulomb is defined in terms of the SI unit for electric current, the ampere, abbreviated A[named after Andre-Marie Ampere (1775 - 1836)].


[named after Andre-Marie Ampere (1775 - 1836)].� The ampere is a basic SI unit like the meter, the second, and the kilogram.

� The unit of charge is defined as

1 C = 1 A s

Charge of an ElectronCharge of an ElectronCharge of an ElectronCharge of an ElectronCharge of an ElectronCharge of an ElectronCharge of an ElectronCharge of an Electron

� We can define the unit of charge in terms of the charge of one electron

• An electron is an elementary particle with charge q = -e where

• e = 1.602 · 10-19 C

• A proton is a particle with q = +e


• A proton is a particle with q = +e

e = 1.602 · 10-19 C

Coulomb of ChargeCoulomb of ChargeCoulomb of ChargeCoulomb of ChargeCoulomb of ChargeCoulomb of ChargeCoulomb of ChargeCoulomb of Charge

� A full coulomb is a very large amount of charge!• A lightning discharge can contain 10’s of coulombs

� The number of electrons required to produce 1 coulomb of charge is

Ne =1 C

1.602 ⋅10-19 C= 6.24 ⋅1018


� Because a coulomb is a large amount of charge, everyday examples of static electricity typically involve• 1 microcoulomb = 1 µC = 10-6 C

• 1 nanocoulomb = 1 nC = 10-9 C

• 1 picocoulomb = 1 pC = 10-12 C

1.602 ⋅10 C

Charge ConservationCharge ConservationCharge ConservationCharge ConservationCharge ConservationCharge ConservationCharge ConservationCharge Conservation

� Benjamin Franklin (1706 - 1790) introduced the idea of positive and negative charge (amber or plastic is negative).

� Franklin also proposed that electric charge is conserved.

� For example,when a plastic rod is charged by rubbing it with a fur, charge is neither created nor destroyed, but instead electrons are transferred to the rod leaving a net positive charge on the fur.


positive charge on the fur.

� Law of charge conservation

• The total charge of an isolated system is strictly conserved.

� This law adds to our list of conservation laws: conservation of energy, conservation of momentum, and conservation of angular momentum

The total charge is constant.

Example Example Example Example ---- Net ChargeNet ChargeNet ChargeNet ChargeExample Example Example Example ---- Net ChargeNet ChargeNet ChargeNet Charge

� Supongamos que queremos crear una carga positiva de 10 µ C en un bloque de cobre de 2,00 kg de masa. ¿Qué fracción en % de electrones deberíamos quitar en el bloque de cobre ?

The atomic weight of copper is 63.55 grams per mole

The number of copper atoms is

Natom

=2.00 kg( ) 6.02 ⋅1023 atoms/mole( )

0.06355 kg/mole= 1.89 ⋅1025


Natom

=0.06355 kg/mole

= 1.89 ⋅10

25 26

,

The atomic number of copper is 29, which means there are 29 electrons per atom

29 29 1.89 10 5.49 10e atom atomN N= ⋅ = ⋅ ⋅ = ⋅

The number of electrons in 10 µC is

Ne,10µC =10 ⋅10−6 C

1.602 ⋅10-19 C= 6.24 ⋅1013

13,10 C -11

26

,

Percentage removed is

6.24 10% 100 100 1.14 10 %

5.49 10

e

e atom

N

N

µ ⋅= ⋅ = ⋅ = ⋅

⋅

� Insulators and ConductorsInsulators and ConductorsInsulators and ConductorsInsulators and Conductors

� SuperconductorsSuperconductorsSuperconductorsSuperconductors

� SemiconductorsSemiconductorsSemiconductorsSemiconductors


Summary …Summary …Summary …Summary …Summary …Summary …Summary …Summary …

� There are two kinds of electric charge – positive and negative.

� Law of Charges• Like charges repel and opposite charges attract

� The unit of charge is the coulomb defined as1 C = 1 A•s


• 1 C = 1 A•s

� Law of charge conservation• The total charge of an isolated system is strictly conserved.

Electrostatic ChargingElectrostatic ChargingElectrostatic ChargingElectrostatic ChargingElectrostatic ChargingElectrostatic ChargingElectrostatic ChargingElectrostatic Charging

� There are two ways to charge an object• Conduction

• Induction


Charging by ConductionCharging by ConductionCharging by ConductionCharging by ConductionCharging by ConductionCharging by ConductionCharging by ConductionCharging by Conduction

Electroscope

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


We brought charge onto the electroscope through contact with the electrode.

InductionInductionInductionInductionInductionInductionInductionInduction

The presence of the positively charged rod leads to a redistribution

- - - - -- - - -

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


leads to a redistribution of charge (a kind of polarization).

It pulls electrons up to the electrode.

The presence of the positively charged rod leads to a redistribution of charge.

Grounding pushes

Charging by InductionCharging by InductionCharging by InductionCharging by InductionCharging by InductionCharging by InductionCharging by InductionCharging by Induction

- - - - -- - - -

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


Grounding pushes positive charge to Earth (or rather pulls electrons from Earth!) leaving the electroscope negative.

ground

Electric Force Electric Force Electric Force Electric Force ---- Coulomb’s LawCoulomb’s LawCoulomb’s LawCoulomb’s LawElectric Force Electric Force Electric Force Electric Force ---- Coulomb’s LawCoulomb’s LawCoulomb’s LawCoulomb’s Law

� Consider two electric charges: q1 and q2

� The electric force F between these two charges separated by a distance r is given by Coulomb’s Law

1 2

2

kq qF

r=


� The constant k is called Coulomb’s constant and is given by

k = 8.99 ⋅109 Nm2 /C2

Coulomb’s Law (2)Coulomb’s Law (2)Coulomb’s Law (2)Coulomb’s Law (2)Coulomb’s Law (2)Coulomb’s Law (2)Coulomb’s Law (2)Coulomb’s Law (2)

� The coulomb constant is also written as

� ε0 is the

k =1

4π ε0 where ε0 = 8.85 ⋅10−12

C2

Nm2


� ε0 is the “electric permittivity of vacuum”

• A fundamental constant of nature

F =1

4πε0

q1q 2

r2

Example: Force between Two ChargesExample: Force between Two ChargesExample: Force between Two ChargesExample: Force between Two ChargesExample: Force between Two ChargesExample: Force between Two ChargesExample: Force between Two ChargesExample: Force between Two Charges

� What is the force between two 1 C charges 1 meter apart?

F =1

4πε0

q1q 2

r2


( )

29

22

N m 1 C 1 C8.99 10

C 1 mF

⋅ ⋅= ⋅

98.99 10 N

which is the weight of 450 Space Shuttles at launch

= ⋅

Electric ForceElectric ForceElectric ForceElectric ForceElectric ForceElectric ForceElectric ForceElectric Force

� The electric force is given by

� The electric force, unlike the gravitational force, can be positive or negative• If the charges have opposite signs, the force is negative• Attractive

F = kq1q2

r2

+ -


• Attractive• If the charges have the same sign, the force is positive• Repulsive

+

-

+

-

-

+

Electric Force VectorElectric Force VectorElectric Force VectorElectric Force VectorElectric Force VectorElectric Force VectorElectric Force VectorElectric Force Vector

Electric force in vector form

x

y

q1

q2

r2

r1

r2 1

2 1ˆ

r r r

r rrr

r r

= −

−= =

r r r

r rr


rF2 = k

q1q2

r2r̂

x r r

rF1 = k

q1q2

r2−r̂( )

Fuerza sobre q2 Fuerza sobre q1

Superposition PrincipleSuperposition PrincipleSuperposition PrincipleSuperposition PrincipleSuperposition PrincipleSuperposition PrincipleSuperposition PrincipleSuperposition Principle

� La fuerza neta que actúa sobre cualquier carga es la suma vectorial de las fuerzas debidas a las cargas restantes de la distribución

1, 1,2 1,3 1,

...

net nF F F F

F F F F

= + + +

= + + +

r r r rL


1 1,2, 1,3, 1, ,

1 1,2, 1,3, 1, ,

1 1,2, 1,3, 1, ,

...

...

...

x x x n x

y y y n y

z z z n z

F F F F

F F F F

F F F F

= + + +

= + + +

= + + +

Example Example Example Example ---- The Helium NucleusThe Helium NucleusThe Helium NucleusThe Helium NucleusExample Example Example Example ---- The Helium NucleusThe Helium NucleusThe Helium NucleusThe Helium Nucleus

� El núcleo de un átomo de helio tiene dos protones y dos neutrones. ¿Cuál es la magnitud de la fuerza eléctrica entre los dos protones en el núcleo de helio?

Each proton has charge q = 1.602 ⋅10−19 C

The distance between the two protons is approximately 2.0 ⋅10-15 m


Each proton has charge q = 1.602 ⋅10−19 C

The force is given by

F = kq1q2

r 2

= 8.99 ⋅109 N ⋅m2

C2

1.602 ⋅10−19 C( )

2

2.0 ⋅10−15 m( )2

= 58 N

Example Example Example Example ---- The Helium Nucleus (2)The Helium Nucleus (2)The Helium Nucleus (2)The Helium Nucleus (2)Example Example Example Example ---- The Helium Nucleus (2)The Helium Nucleus (2)The Helium Nucleus (2)The Helium Nucleus (2)

� ¿Qué pasa si la distancia r entre los protones se duplica?

F2r = kq1q2

2r( )2= kq1q2

4r2 with Fr = k

q1q2

r( )2

F2r =1

4Fr


Ley del inverso del cuadrado: Si la distancia se duplica entonces la fuerza se reduce por un factor 4.

F2r =4Fr

F2r =1

458 N( )= 14.5 N

� Considera dos cargas colocadas en el eje x

• q1 = 0.15 µC x1 = 0.0 m

x1 x2

Clicker Quiz: Equilibrium PositionClicker Quiz: Equilibrium PositionClicker Quiz: Equilibrium PositionClicker Quiz: Equilibrium PositionClicker Quiz: Equilibrium PositionClicker Quiz: Equilibrium PositionClicker Quiz: Equilibrium PositionClicker Quiz: Equilibrium Position

A B C


• q1 = 0.15 µC x1 = 0.0 m• q2 = 0.35 µC x2 = 0.40 m

� ¿Dónde tenemos que poner una tercera carga para que esa carga pueda estar en equilibrio?

� : to the left of charge 1� : in the middle between the two charges� : to the right of charge 2

A

B

C

Clicker Quiz: Equilibrium Position (2)Clicker Quiz: Equilibrium Position (2)Clicker Quiz: Equilibrium Position (2)Clicker Quiz: Equilibrium Position (2)Clicker Quiz: Equilibrium Position (2)Clicker Quiz: Equilibrium Position (2)Clicker Quiz: Equilibrium Position (2)Clicker Quiz: Equilibrium Position (2)

� A: x3<x1• Here the forces from q1 and q2 will always point in the same direction (to the left for a positive test charge)• No equilibrium

x1 x2


• No equilibrium

� C: x2<x3• Here the forces from q1 and q2 will always point in the same direction (to the right for a positive test charge)• No equilibrium

Clicker Quiz: Equilibrium Position (3)Clicker Quiz: Equilibrium Position (3)Clicker Quiz: Equilibrium Position (3)Clicker Quiz: Equilibrium Position (3)Clicker Quiz: Equilibrium Position (3)Clicker Quiz: Equilibrium Position (3)Clicker Quiz: Equilibrium Position (3)Clicker Quiz: Equilibrium Position (3)

� B: x1 < x3 < x2• Here the forces from q1 and q2 can balance• q1 = 0.15 µC x1 = 0.0 m

x1 x2

q3

1 3 3 2 q q q q

k k=


• q1 = 0.15 µC x1 = 0.0 m• q2 = 0.35 µC x2 = 0.40 m

1 3 3 2

2 2

3 1 2 3

( ) ( )

q q q qk kx x x x

=− −

q1

(x3 − x1)2

=q2

(x2 − x3)2⇒

q1(x2 − x3)2 = q2 (x3 − x1)

2 ⇒

q1 (x2 − x3) = q2 (x3 − x1)⇒

x3 =q1x2 + q2 x1

q1 + q2

x3 =q1x2 + q2 x1

q1 + q2=

0.15 µC ⋅ (0.4 m)

0.15 µC + 0.35 µC= 0.16 m

Example Example Example Example ---- Charged Balls (1)Charged Balls (1)Charged Balls (1)Charged Balls (1)Example Example Example Example ---- Charged Balls (1)Charged Balls (1)Charged Balls (1)Charged Balls (1)

� Consideremos dos bolas idénticas cargadas que cuelgan del techo por hilos de igual longitud ℓ = 1,5 m (en equilibrio). Cada bola tiene una carga de 25 µ C. Las bolas cuelgan separadas un ángulo 2θ , θ = 25° con respecto a la vertical. ¿Cuál es la masa de las bolas?

� Think� Three forces act on each ball:


� Three forces act on each ball:• Coulomb force Fc, gravity Fg, and the tension of the string T

� The coulomb force is horizontal and must be repulsive to keep the balls apart

� The gravitational force points down� The tension T is in the direction of the string� The balls are in equilibrium, which means the three forces are

balanced


� Sketch� Add the three forces to the drawing� Define distance d between the balls� Make a separate free body diagram for one of the charged

balls� Define x-y coordinate system



� Research� The condition of static equilibrium tells us

that the sum of the x-components of thethree forces acting on the ball must equalzero and the sum of y-components of theseforces must equal zero

� The sum of the x-components of the forces is

• T is magnitude of the string tension• θ is the angle of the string relative to the vertical

T sinθ − F

c= 0


• θ is the angle of the string relative to the vertical• FC is the magnitude of the Coulomb force

� The sum of the y-components of the forces is

� The force of gravity, Fg, is just the weight of the charged ball

• m is the mass of the charged ball

T cosθ − F

g= 0

Fg

= mg


� The electric force between the two balls is

• d is the distance between the two balls

� We can express d in terms of ℓ and θ

Fc

= kq2

d 2

/2sin

dθ =


� We can then rewrite the electric force interms of θ and ℓ

/2sin

dθ =

l

( )

2 2

2 2 24 sin2 sinc

q qF k k

θθ= =

ll

Example Example Example Example ---- Charged Balls (5)Charged Balls (5)Charged Balls (5)Charged Balls (5)Example Example Example Example ---- Charged Balls (5)Charged Balls (5)Charged Balls (5)Charged Balls (5)� Simplify

� Thus we eliminate the string tension and get

� Putting in the force of gravity and the electric force,

T sinθ = Fc

T cosθ = Fg

tanθ =Fc

Fg


� Putting in the force of gravity and the electric force,

� Solving for m, we get

2

22 2

2 2

4 sintan4 sin

qk

kq

mg mg

θθθ

= =l

l

2

2 24 sin tan

kqm

g θ θ=

l


� Calculate� Putting in our numerical values we obtain

� Round

� We report our result to three significant figures

m =8.99 ⋅109 N ⋅m2 /C2( ) 25.0 µC( )

2

4 ⋅ 9.81 m/s2( )1.50 m( )2

sin2 25.0°( )tan 25.0°( )= 0.764116 kg


� We report our result to three significant figures

� Double-check

� Para comprobar nuestra respuesta, hacemos la aproximación senθ ≈tanθ ≈ θ y cosθ ≈ 1 , de modo que T ≈ mg y las componentes x de las fuerzas son

m = 0.764 kg

( )

2 2

22sin

2c

q qT mg F k k

dθ θ

θ≈ = = ≈

l


� The mass of the ball in our double-check is then

� Which is close to our answer

( )( )

( )( ) ( )

29 2 22

2 32 3 2

8.99 10 N m /C 25.0 C0.768 kg

4 4 9.81 m/s 1.50 m 0.436 rad

kqm

g

µ

θ

⋅ ⋅= = =

⋅l


� Which is close to our answer

m = 0.764 kg

Example Example Example Example ---- Forces between ElectronsForces between ElectronsForces between ElectronsForces between ElectronsExample Example Example Example ---- Forces between ElectronsForces between ElectronsForces between ElectronsForces between Electrons

� What is relative strength of the electric force compared with the force of gravity for two electrons?

Felectric

Fgravity=kqe

2

Gme2

=(8.99 ⋅109 N ⋅m2 / C2 )(1.602 ⋅10−19 C)2

(6.67 ⋅10-11 N ⋅m2 /kg2 )(9.109 ⋅10-31 kg)2= 4.2 ⋅1042

Felectric = kqe2

r2

Fgravity = Gme2

r2


� Gravity is irrelevant for atomic and subatomic processes – the electric force is much much stronger.

� But sometimes gravity is most important; e.g., the motion of the planets. Why?

r

Example Example Example Example ---- Four ChargesFour ChargesFour ChargesFour ChargesExample Example Example Example ---- Four ChargesFour ChargesFour ChargesFour Charges

� Considere cuatro cargas colocadas en las esquinas de un cuadrado con lados de longitud 1,25 m, como se muestra a la derecha. ¿Cuál es la magnitud de la fuerza eléctrica sobre q4 resultante de la fuerza eléctrica de los restantes tres cargas?

2Set up x-y coordinate system with its origin located at q y


x-direction

Fx = kq1q4

d2+ k

q2q4

2d( )2cos45° =

kq4

d 2q1 +

q2

2cos45°

y-direction

Fy = kq2q4

2d( )2sin 45° + k

q3q4

d 2=kq4

d 2q2

2sin 45° + q3

x

Example Example Example Example ---- Four Charges (2)Four Charges (2)Four Charges (2)Four Charges (2)Example Example Example Example ---- Four Charges (2)Four Charges (2)Four Charges (2)Four Charges (2)

F = Fx2 + Fy

2

F =kq4

d2q1 +

q2

2cos45°

2

+kq4

d2q2

2sin 45° + q3

2

F =kq4

d2q1 +

q2

2cos45°

2

+q2

2sin 45° + q3

2

y


q2

2sin 45° =

q2

2cos45° =

2.50 µC

2 ⋅ 2= 0.884 µC

F =8.99 ⋅109( ) 4.50 µC( )

1.25 m( )21.50 µC + 0.884 µC( )2 + 0.884 µC − 3.50 µC( )2

F = 0.0916 N

x

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Electricity and Magnetism - personales.unican.espersonales.unican.es/lopezqm/FBE/elmenu/teoria/FBE=Tema VIII_20121… · Fundamental Forces of Nature The force of gravity has been