1
ElectrostaticsElectrostatics
Physics for Scientists & Engineers 2, Chapter 21 1
ElectrostaticsElectrostatics Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 2
Physics 184Physics 18421.1 Electromagnetism21.1 Electromagnetism
Electricity 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 is derived from the Greek word for amber
Physics for Scientists & Engineers 2, Chapter 21 3
from the Greek word for amber
• 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
Fundamental Forces of NatureFundamental Forces of Nature
Physics for Scientists & Engineers 2, Chapter 21 4
The 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
Physics for Scientists & Engineers 2, Chapter 21 5
(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
Gravitational and Electric ForcesGravitational and Electric Forces
For gravity we defined a gravitational force
and a gravitational potential
F(r)=G
m1m2
r2
Physics for Scientists & Engineers 2, Chapter 21 6
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)=-G
m1m2
r
2
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 7
Physics 184Physics 18421.2a Electric Charge21.2a Electric Charge
Electric 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
• A spark occurs when the built-up charge discharges through
Physics for Scientists & Engineers 2, Chapter 21 8
A spark occurs when the built up charge discharges through the metal of the door knob.
Similar phenomenon involving wind, rain and ice produces lightning
Electric ChargeElectric Charge
Normally objects around us do not seem to carry a net charge
They have equal amounts of positive and negative charge and thus are electrically neutral• Negative charge means an excess of electrons• Positive charge means a deficit of electrons
If a plastic rod is rubbed with fur, the rod will become
Physics for Scientists & Engineers 2, Chapter 21 9
p ,charged• If two charged plastic rods together, they will repel each
other If a glass rod is rubbed 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
This result leads to the Law of Electric Charges
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
Law of Electric ChargesLaw of Electric Charges
Like charges repel and opposite charges attract.Like charges repel and opposite charges attract.Like charges repel and opposite charges attract.
electric current, the ampere, abbreviated A• 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
Physics for Scientists & Engineers 2, Chapter 21 10
1 C = 1 A s1 C = 1 A s
Charge 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 coulomb is a large amount of charge
Typically we deal with smaller amounts of charge
Physics for Scientists & Engineers 2, Chapter 21 11
Typically we deal with smaller amounts of charge
• 1 microcoulomb = 1 μC = 1.0·10-6 C
• 1 nanocoulomb = 1 nC = 1.0·10-9 C
• 1 picocoulomb = 1 pC = 1.0·10-12 C
The number of electrons required to make a coulomb is
• Ne = 1 C/(1.602·10-19 C per electron) = 6.242·1018 electrons
Charge 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
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
Physics for Scientists & Engineers 2, Chapter 21 12
transferred to the rod leaving a net positive charge on the fur
Law of Charge Conservation
This law adds to our list of conservation laws• Conservation of energy• Conservation of momentum• Conservation of angular momentum
The total charge is constant.The total charge is constant.
3
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 13
Physics 184Physics 184Example: Muscle TwitchExample: Muscle Twitch
Muscle TwitchMuscle Twitch
PROBLEM
A current of 5.0 mA is enough to make your muscles twitch. How many electrons flow through your skin if you are exposed to such a current for 10.0 s?
SOLUTION
3 C
Physics for Scientists & Engineers 2, Chapter 21 14
3 C5.0 mA 5.0 10
s-= ⋅
( ) 3 C10.0 s 5.0 10 0.050 C
s-æ ö÷ç ⋅ =÷ç ÷çè ø
1719
1 electron0.050 C 3.1 10 electrons
1.602 10 C- = ⋅⋅
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 15
Physics 184Physics 18421.2b Elementary Charge21.2b Elementary Charge
Elementary ChargeElementary Charge
Electric charge is quantized
The smallest charge observable is the charge of anelectron
Established by Robert Millikan (1868 - 1953) in his famous
Physics for Scientists & Engineers 2, Chapter 21 16
(1868 - 1953) in his famousoil drop experiment
In everyday life, we don’tnotice that charge is quantized because most electrical phenomena involve a large number of electrons
Structure of AtomsStructure of Atoms
Atoms are electrically neutral
Atoms are composed of a positively charged atomic nucleus surrounded by negative electrons
Physics for Scientists & Engineers 2, Chapter 21 17
electrons The atomic nucleus is
composed of positively charge protons and electrically neutral neutrons
The number of protons is the same as the number of electrons
Description of AtomsDescription of Atoms
Atomic number = ZMass number = A# electrons = Z (charge = -Ze)# protons = Z (charge = +Ze)# neutrons = N = A – ZFor example, 12C has
Physics for Scientists & Engineers 2, Chapter 21 18
6 protons6 neutrons6 electrons
Atomic mass = ZMp + NMn
ZMe – binding energy/c2
Atomic mass ≈ AMp
4
ParticlesParticles
The electron is an elementary particle
The proton is composed of chargedparticles called quarks held together byuncharged particles call gluons• Quarks have charge ±(1/3)e and ±(2/3)e
• Quarks have never been observed directlyQuarks have never been observed directly
The proton is composed of two up quarkseach with charge +(2/3)e and one downquark with charge –(1/3)e
The neutron is composed of one up quark with charge +(2/3)e and twodown quarks with charge –(1/3)e
Physics for Scientists & Engineers 2, Chapter 21 19
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 20
Physics 184Physics 184Example: Net ChargeExample: Net Charge
Net ChargeNet Charge
PROBLEM
Suppose we want to create a positive charge of 10.0 μC on a block of copper metal with mass 2.00 kg. What fraction of the electrons in the copper block would we remove?
SOLUTION
Physics for Scientists & Engineers 2, Chapter 21 21
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
Net ChargeNet Charge
eq eN=
The atomic number of copper is 29, which means
there are 29 electrons per atom
Ne = 29⋅Natom = 29⋅1.89⋅1025 =5.49⋅1026
The number of electrons in 10 C is thus
Physics for Scientists & Engineers 2, Chapter 21 22
13e -11
26e
Percentage removed is
6.24 10% 100 100 1.14 10 %
5.49 10
N
N ⋅
= ⋅ = ⋅ = ⋅⋅
613
e -19
10 10 C6.24 10
1.602 10 C
qN
e
-⋅= = = ⋅
⋅
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 23
Physics 184Physics 18421.3 Insulators and Conductors21.3 Insulators and Conductors
Insulators and ConductorsInsulators and Conductors
The electronic structure of materials determines their ability to conduct electricity• “Conducting electricity” means the transport of electrons
Materials that conduct electricity well are called conductors• Electrons can move freely (some of the electrons)
M t l
Physics for Scientists & Engineers 2, Chapter 21 24
• Metals• Water with dissolved materials
Materials that conduct electricity poorly are called insulators• Electrons cannot move freely
• Glass• Plastic• Cloth• Pure water
5
SemiconductorsSemiconductors Semiconductors are materials that can be switched
between being an insulator and being a conductor
Semiconductors are the backbone of modern electronics and computers
Physics for Scientists & Engineers 2, Chapter 21 25
Replica of first transistor in 1947
Modern computer chip with millions of transistors
SuperconductorsSuperconductors
Superconductors are materials that have no resistance to the conduction of electricity as opposed to normal conductors that conduct electricity well but with some losses
A typical superconductor is a niobium-titanium alloy that must be kept near the temperature of liquid
Physics for Scientists & Engineers 2, Chapter 21 26
that must be kept near the temperature of liquid helium (4.2 K)
During the last 20 years, high temperature superconductors have been developed that operated at the temperature of liquid nitrogen (77.3 K)
Material that are superconductors at room temperature would be very useful
Applications of SuperconductorsApplications of Superconductors
The main application of superconductors is to produce electromagnets made with superconducting wire
Examples include• Magnetic resonance imaging (MRI) machines• Particle accelerators
• MSU’s K1200 Superconducting Cyclotron
Physics for Scientists & Engineers 2, Chapter 21 27
• MSU’s upcoming superconducting LINAC (linear accelerator) for FRIB
• Tevatron at Fermilab near Chicago, Illinois• RHIC at Brookhaven National Laboratory on Long Island,
NY• LHC at CERN near Geneva, Switzerland
Magnetic Resonance Imaging - MRIMagnetic Resonance Imaging - MRI
MRI stands for nuclear magnetic resonance imaging
MRI produces high quality images of living tissue without causing any damage
The quality of an MRI image (signal to noise) is proportional to
Magnetic Field = 1.5 T
Yue Cao, Stephen Whalen, Jie Huang, Kevin L. Berger, and Mark C. DeLanHuman Brain Mapping 20:82–90(2003). (MSU Radiology)
Physics for Scientists & Engineers 2, Chapter 21 28
(signal-to-noise) is proportional to the the magnitude of the magnetic field• High field means high quality images
Superconducting magnets can produce up to four times the magnetic field of a room-temperature magnet
Magnetic Field = 3.0 T
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 29
Physics 184Physics 18421.4 21.4 Electrostatic ChargingElectrostatic Charging
Electrostatic ChargingElectrostatic Charging
Giving a static charge to an object is calledelectrostatic charging
We will approach our study of electrostatic charging through a series of simple experiments
A power supply or a battery can provide positive and negative charge
Physics for Scientists & Engineers 2, Chapter 21 30
negative charge
Insulating paddles can be positively or negatively charged
We will make a conducting connection to the Earth• We call this connection grounding
• The Earth is a nearly infinite reservoir of charge and neutralizes electrically charged objects connected to it
6
ElectroscopeElectroscope
An electroscope gives an observable response when it is charged
This electroscope has two conductors that are initially vertical and touching when thevertical and touching when the electroscope is uncharged
When the electroscope is charged, the hinged conductor will move away from the fixed conductor
Note that we cannot tell the sign of the charge
Physics for Scientists & Engineers 2, Chapter 21 31
Inducing a Negative ChargeInducing a Negative Charge
If we bring a negatively charged paddle near an electroscope, the electrons are repelled from the ball, inducing a negative charge on the conductors
Physics for Scientists & Engineers 2, Chapter 21 32
Charging by ContactCharging by Contact
If the negatively charged paddle touches the electroscope, electrons will flow from the paddle to the conductors, producing a net negative charge
Physics for Scientists & Engineers 2, Chapter 21 33
Positive Charging by InductionPositive Charging by Induction
We can create a positive charge• Starting with an uncharged electroscope
• Bringing a negatively charged paddle close to the electroscope
• Connecting a ground to the electroscope
• Removing the ground
• Taking the paddle away
Physics for Scientists & Engineers 2, Chapter 21 34
QuestionQuestion
Two lightweight metal spheres are suspended near each other from insulating threads. One sphere has a net charge; the other sphere has no net charge. The spheres will
A. attract each other.
B exert no net electrostatic force on each otherB. exert no net electrostatic force on each other.
C. repel each other.
D. do any of these things depending on the sign of the charge on the one sphere.
Physics for Scientists & Engineers 2, Chapter 21 35
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 36
Physics 184Physics 18421.5a Electrostatic Force 21.5a Electrostatic Force –– Coulomb’s LawCoulomb’s Law
7
Electrostatic Force – Coulomb’s LawElectrostatic Force – Coulomb’s Law The law of electric charges is evidence of a force
between any two charges at rest
Experiments show that for the electrostatic force exerted by charge 2 q2 on charge 1 q1, the force on q1
points toward q2 if the charges have opposite signs and away from q2 if the charges have like signs
Physics for Scientists & Engineers 2, Chapter 21 37
Electrostatic Force – Coulomb’s LawElectrostatic Force – Coulomb’s Law
Coulomb’s Law gives the magnitude of this force as
k is Coulomb’s constant given by
1 2
2
1 2
1 2
and are electric charges
is the distance between the charges
q qF k
rq q
r r r
=
= -
We can relate Coulomb’s constant to theelectric permittivity of free space ε0
Physics for Scientists & Engineers 2, Chapter 21 38
k = 8.99⋅109
N m2
C2
k =
1
40
0 = 8.85⋅10-12 C2
N m2
Electrostatic Force VectorElectrostatic Force Vector
Coulomb’s Law can be written in vector form as
( )1 22 1 2 13
q qF k r r
r
= - - 1 2
212ˆ
q qk r
r= -
Physics for Scientists & Engineers 2, Chapter 21 39
Superposition PrincipleSuperposition Principle
Consider the force exerted on charge q3 by two other charges q1 and q2
Physics for Scientists & Engineers 2, Chapter 21 40
( )1 3
1
1 3 23
ˆ q q
k xx x
F
= -- ( )
22 3
32
3 2
ˆ q q
k xx x
F
= --
1 3net 23 3F FF
= +
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 41
Physics 184Physics 184Example: Force between Two ChargesExample: Force between Two Charges
Force between Two Charges Force between Two Charges
You place two charges a distance r apart. Then you double each charge and double the distance between the charges. How does the force between the two charges change?
A. The new force is twice as large.
B The new force is half as largeB. The new force is half as large.
C. The new force is four times as large.
D. The new force is four times smaller.
E. The new force is the same.
Physics for Scientists & Engineers 2, Chapter 21 42
8
Force between Two Charges (2)Force between Two Charges (2)
PROBLEM
What is the magnitude of the force between two 1.00 C charges 1.00 m apart?
SOLUTION
1 2
2
q qF k
r=
Physics for Scientists & Engineers 2, Chapter 21 43
This force is approximately the same as 450 loaded Space Shuttles!
( )( )
229
22
9
1.00 CN m8.99 10
C 1.00 m
8.99 10 N
F
F
æ ö÷ç ÷= ⋅ç ÷ç ÷çè ø
= ⋅
r Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 44
Physics 184Physics 184Example: Electrostatic Force inside the AtomExample: Electrostatic Force inside the Atom
Electrostatic Force inside the AtomElectrostatic Force inside the Atom PROBLEM 1
What is the magnitude of the electrostatic force between the two protons in the helium nucleus?
SOLUTION 1
The two neutrons and protons inside the helium nucleus are held together by the strong forcenucleus are held together by the strong force
The two protons each have charge +e and are separated by a distance of 2.0·10-15 m
The electrostatic force of repulsion between the two protons is
Physics for Scientists & Engineers 2, Chapter 21 45
F= kq1q2
r2= 8.99⋅109
N m2
C2
æ
èççç
ö
ø÷÷÷÷
1.60⋅10-19 C( )2
2.0⋅10-15 m( )2 =58 N
Electrostatic Force inside the Atom(2)Electrostatic Force inside the Atom(2) PROBLEM 2
What is the magnitude of the electrostatic force between a gold nucleus and an electron of the gold atom in an orbit with radius 4.88·10–12 m?
SOLUTION 2
The attractive force between the negative electron and gthe positive gold nucleus is
Physics for Scientists & Engineers 2, Chapter 21 46
F= kq1q2
r2= k
e Ze( )r2
= kZe2
r2
F= 8.99⋅109 N m2
C2
æ
èççç
ö
ø÷÷÷÷
79( ) 1.60⋅10-19 C( )2
4.88⋅10-12 m( )2 = 7.63⋅10-4 N
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 47
Physics 184Physics 184Example: Equilibrium PositionExample: Equilibrium Position
Equilibrium PositionEquilibrium Position
PROBLEM
Two charged particles are placed as shown in the figure:q1 = 0.15 μC is located at the origin, and q2 = 0.35 μC is located on the positive x-axis at x2 = 0.40 m. Where should a third charged particle, q3, be placed to be at an equilibrium point (the forces on it sum to zero)?
Physics for Scientists & Engineers 2, Chapter 21 48
9
Equilibrium PositionEquilibrium Position
SOLUTION
Let’s figure out where not to put the third charge
If the charge is placed anywhere off the x-axis, there will be a component of the force toward or away from the x-axis• Third charge must lie on the x-axis
Look at three regionsx < 0, 0 < x < x2, x > x2
Physics for Scientists & Engineers 2, Chapter 21 49
Equilibrium PositionEquilibrium Position
If x < 0 or x > x2 then the force on the third charge from the the other two charges would be in the same direction regardless of the sign of the third charge• No equilibrium possible
If 0 < x < x2 then the force on charge three from the other two charges would be in the opposite direction regardless of the sign of the third charge• Equilibrium possible
Physics for Scientists & Engineers 2, Chapter 21 50
Equilibrium PositionEquilibrium Position
rF1 3 =
rF2 3 k
q1q32 = k
q2q32
Physics for Scientists & Engineers 2, Chapter 21 51
F13 F23 kx3 -x1( )2 k
x2 -x3( )2
q1
x3 -x1( )2 =q2
x3 -x2( )2 q1 x2 -x3( )2 = q2 x3 -x1( )2
q1 x2 -x3( )= q2 x3 -x1( ) q1 x2 - q1 x3 = q2 x3 - q2 x1
q1 x3 + q2 x3 = q1 x2 + q2 x1
Equilibrium PositionEquilibrium Position
q x + q x
Physics for Scientists & Engineers 2, Chapter 21 52
x3 =q1 x2 + q2 x1
q1 + q2
x3 =0.15⋅10-6 C 0.40 m( )
0.15⋅10-6 C + 0.35⋅10-6 Cx3 = 0.16 m
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 53
Physics 184Physics 184Solved Problem 21.1Solved Problem 21.1
Charged BallsCharged Balls
Charged BallsCharged Balls
PROBLEM Two identical charged balls
hang from the ceiling by insulated ropes of equal length, = 1.50 mA h 25 0 C i li d A charge q = 25.0 μC is applied to each ball
Then the two balls hang at rest, and each supporting rope has an angle of 25.0° with respect to the vertical
What is the mass of each ball?
Physics for Scientists & Engineers 2, Chapter 21 54
10
Charged BallsCharged Balls
SOLUTIONTHINK
Each ball has three forces acting on it• Force of gravity• Repulsive electrostatic force• Tension in the supporting rope
The forces must sum to zero
Physics for Scientists & Engineers 2, Chapter 21 55
e esin 0 sinT F T F - = =
The forces must sum to zeroSKETCH
A free body diagram of one of theballs is shownRESEARCH
The sum of the x-components of the forces gives us
Charged BallsCharged Balls
The sum of the y-components of the forces gives us
The electrostatic force is given by
The force of gravity is given by
g gcos 0 cosT F T F - = =
2
e 2
qF k
d=
F mg=
The distance between the balls is given by
The electrostatic force between the balls is
Physics for Scientists & Engineers 2, Chapter 21 56
gF mg=
/ 2sin
2
d d
= =
( )
2 2
e 2 2 24 sin2 sin
q qF k k
= =
Charged BallsCharged Balls
SIMPLIFY We divide two force component equations to get
Substitute in our equations for Fe and Fg
e e
g g
sin tan =
cos
T F F
T F F
=
2q
CALCULATE
Physics for Scientists & Engineers 2, Chapter 21 57
22 2
2 24 sintan =
4 sin tan
qk kq
mmg g
=
( )
( )( )
229 6
2
22 2
N m8.99 10 25.0 10 C
C0.764116 kg
4 9.81 m/s 1.50 m sin 25.0 tan 25.0m
-æ ö÷ç ÷⋅ ⋅ç ÷ç ÷çè ø
= =
Charged BallsCharged Balls
ROUND
DOUBLE-CHECK To double-check our results, let’s make the small angle
approximation that sinθ ≈ tanθ ≈ θ and cosθ ≈ 1
0.764 kgm =
2 2
Physics for Scientists & Engineers 2, Chapter 21 58
( )
( )
( ) ( )
2 2
e 22
229 6
22
2 32 3
sin2
N m8.99 10 25.0 10 C
C0.768 kg
4 4 1.50 m 0.436 rad
q qT mg F k k
d
kqm
g g
-
» = = »
æ ö÷ç ÷⋅ ⋅ç ÷ç ÷çè ø= = =
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 59
Physics 184Physics 184Solved Solved Problem 21.2Problem 21.2
Four ChargesFour Charges
Four ChargesFour ChargesPROBLEM
Consider four charges placed at the corners of a square with sides of length 1.25 m as shown on the right
What is the magnitude of the electric force on q4 resulting from the electric force from the remaining three charges?
Physics for Scientists & Engineers 2, Chapter 21 60
charges?
SOLUTIONTHINK
The electrostatic force on q4 is the vector sum of the forces resulting from its interactions with the other three charges
11
Four ChargesFour Charges
We need to determine the individual force components in each spatial direction and add those to find the components of the net force vector
SKETCH
Physics for Scientists & Engineers 2, Chapter 21 61
Four ChargesFour Charges
RESEARCH
The net force on q4 is the vector sum of the forces
The x-component of these forces is1 4 2 4 3 4, ,F F F
Fx kq1q4
2 kq2q4
2cos45
kq42 q1
q2 cos45
Physics for Scientists & Engineers 2, Chapter 21 62
The y-component of these forces is
The magnitude and angle of the net force are
Fy kq2q4
2d 2sin 45 k
q3q4
d 2 kq4
d 2
q2
2sin 45q3
Fx kd 2 k
2d 2cos45
d 2 q1 2cos45
F Fx2 Fy
2 tan Fy / Fx
Four ChargesFour Charges
SIMPLIFY
Putting in our expressions for the components of the force
Fkq4
d2 q1 q2
2cos45
2
kq4
d 2
q2
2sin 45q3
2
2
2
Physics for Scientists & Engineers 2, Chapter 21 63
For the angle of the force we have
tan1Fy
Fx
tan1
kq4
d 2
q2
2sin 45q3
kq4
d 2 q1 q2
2cos45
tan1
q2
2sin 45q3
q1 q2
2cos45
F
kq4
d 2 q1 q2
2cos45
2
q2
2sin 45q3
2
Four ChargesFour Charges
CALCULATE
Putting in our numerical values, we get
2 2 2.50 Csin 45 cos 45 0.8839 C
2 2 2 2
q q
9 2 2
2 2
2
8.99 10 N m /C 4.50 C1.50 C 0.8839 C 0.8839 C 3.50 C
1.25 mF
Physics for Scientists & Engineers 2, Chapter 21 64
ROUNDF 0.0916 N
47.7
0.09164 NF
1 0.8839 C 3.50 Ctan 47.66
1.50 C 0.8839 C
Four ChargesFour Charges
DOUBLE-CHECK
To double-check our result, we calculate the magnitude of the three forces acting on q4
F14 kq1q4
r142 8.99109 N m2 /C2 1.50 C 4.50 C
1.25 m 2 0.0388 N
F kq2q4 8 99 109 N 2 /C2 2.50 C 4.50 C
0 0324 N
Physics for Scientists & Engineers 2, Chapter 21 65
The angle looks reasonable because the force points downward and to the right
F24 kq2q4
r242 8.99109 N m2 /C2
2 1.25 m 2 0.0324 N
F34 kq3q4
r342 8.99109 N m2 /C2 3.50 C 4.50 C
1.25 m 2 0.0906 N
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 66
Physics 184Physics 18421.5b Electrostatic Applications21.5b Electrostatic Applications
12
Electrostatic PrecipitatorElectrostatic Precipitator
An application of electrostatic charging and electrostatic forces is the cleaning of emissions from coal-fired power plants
An electrostatic precipitatorremoves ash and particulates
Physics for Scientists & Engineers 2, Chapter 21 67
Laser PrinterLaser Printer
A laser printer uses electrostatic forces to print
Physics for Scientists & Engineers 2, Chapter 21 68
Laser PrinterLaser Printer
Any spot the laser strikes onthe drum is discharged, allowing toner to produce an image
Physics for Scientists & Engineers 2, Chapter 21 69
Completely charged drum
Physics 184Physics 184
Physics for Scientists & Engineers 2, Chapter 21 70
Physics 184Physics 18421.6 Coulomb’s Law and Newton’s21.6 Coulomb’s Law and Newton’s
Law of GravitationLaw of Gravitation
Forces between ElectronsForces between Electrons What is relative strength of the electric force
compared with the force of gravity for two electrons?
Fe
Fg
kqe
2
Gme2
(8.99109 N m2 / C2 )(1.6021019 C)2
(6.67 10-11 N m2 /kg2 )(9.10910-31 kg)2 4.21042
Fe = kqe
2
r2
Fg =Gme
2
r2
Physics for Scientists & Engineers 2, Chapter 21 71
Gravity is irrelevant for atomic and subatomic processes – the electric force is much muchstronger
But sometimes gravity is most important; e.g, the motion of the planets
r