1) The document discusses Faraday's law, Coulomb's law, and Gauss's law relating to electric and magnetic fields. It provides explanations and examples of these physical laws.
2) Faraday's law states that a changing magnetic field can induce an electromotive force (emf). The document gives an example problem calculating the current induced in a coil.
3) Coulomb's law describes the electrostatic force of attraction or repulsion between electric charges. The document discusses electric charge and how charges move in conductors versus insulators.
4) Gauss's law relates the electric flux through a closed surface to the electric charge enclosed by the surface. The document gives examples of applying Gauss's law to different gaussian surfaces
3. Area ChangeArea Change
The sliding bar creates an emfThe sliding bar creates an emf
by changing the area in theby changing the area in the
magnetic field.magnetic field.
• Constant magnetic fieldConstant magnetic field
The potential was due to theThe potential was due to the
time rate of change of area.time rate of change of area.
t
AB
vBLV
∆
∆
==∆
4. Field ChangeField Change
An emf can also be generatedAn emf can also be generated
by changing the magnetic field.by changing the magnetic field.
The time rate of change of theThe time rate of change of the
field through a fixed loopfield through a fixed loop
provides the potential.provides the potential.
t
BA
V
∆
∆
=∆
5. Field OrientationField Orientation
The emf depends on theThe emf depends on the
change in field or the changechange in field or the change
in area.in area.
• Area perpendicular to the fieldArea perpendicular to the field
This suggests that the productThis suggests that the product
of the field and areaof the field and area
perpendicular matters.perpendicular matters.
( ) ( )
t
AB
t
BA
V
∆
∆
=
∆
∆
=∆ ⊥ θcos
6. Magnetic FluxMagnetic Flux
The product of the field and area perpendicular to theThe product of the field and area perpendicular to the
field is thefield is the magnetic fluxmagnetic flux..
The magnetic flux is measured in webers.The magnetic flux is measured in webers.
• 1 Wb = 1 T m1 Wb = 1 T m22
The magnetic field can be thought of as a flux density.The magnetic field can be thought of as a flux density.
θcosABM =Φ
A
B MΦ
=
7. Faraday’s LawFaraday’s Law
The flux can be used to getThe flux can be used to get
the induced emf.the induced emf.
• Sign indicates polaritySign indicates polarity
This is Faraday’s Law ofThis is Faraday’s Law of
induction.induction.
For multiple turns the emf isFor multiple turns the emf is
multiplied.multiplied.
• NN turns of wireturns of wire
• NNΦΦ is the flux linkageis the flux linkage
t
M
∆
∆Φ
−=ε
t
N M
∆
∆Φ
−=ε
8. Coil FluxCoil Flux
A circular flat coil has 200A circular flat coil has 200
turns of wire with a totalturns of wire with a total
resistance of 25resistance of 25 ΩΩ and anand an
enclosed area of 100 cmenclosed area of 100 cm22
..
There is a perpendicularThere is a perpendicular
magnetic field of 0.50 T that ismagnetic field of 0.50 T that is
turned off in 200 ms.turned off in 200 ms.
Find the current induced in theFind the current induced in the
coil.coil.
This problem has three parts.This problem has three parts.
To get the current from theTo get the current from the
resistance the voltage isresistance the voltage is
needed.needed.
To get the voltage the flux isTo get the voltage the flux is
needed.needed.
• Flux linkage works, tooFlux linkage works, too
Find the flux first.Find the flux first.
9. Flux to CurrentFlux to Current
The magnetic flux isThe magnetic flux is ΦΦ = BA= BA..
• ΦΦ = (0.50 T)(100 cm= (0.50 T)(100 cm22
))
• ΦΦ = (0.50 T)(0.010 m= (0.50 T)(0.010 m22
))
• ΦΦ = 0.0050 T m= 0.0050 T m22
The change in flux isThe change in flux is
negative since it is turnednegative since it is turned
off.off.
The induced emf isThe induced emf is
• EE == −−NN ∆Φ∆Φ//∆∆tt
• EE = -(200)(-0.0050 Tm= -(200)(-0.0050 Tm22
) /) /
(0.20 s)(0.20 s)
• EE == ∆∆VV = 5.0 V= 5.0 V
The induced current comesThe induced current comes
from Ohm’s Law.from Ohm’s Law.
• I = V/RI = V/R
• II = (5.0 V) / (25= (5.0 V) / (25 ΩΩ))
• II = 0.20 A= 0.20 A
next
10. September 26, 2007September 26, 2007
Gauss’ LawGauss’ Law
We are going to be most interested inWe are going to be most interested in closedclosed
surfaces, in which case the outward directionsurfaces, in which case the outward direction
becomes self-evident.becomes self-evident.
We can ask, what is the electric flux out of such aWe can ask, what is the electric flux out of such a
closed surface? Just integrate over the closedclosed surface? Just integrate over the closed
surface:surface:
The symbol has a little circle to indicate thatThe symbol has a little circle to indicate that
the integral is over a closed surface.the integral is over a closed surface.
The closed surface is called aThe closed surface is called a gaussian surfacegaussian surface,,
because such surfaces are used by Gauss’ Law,because such surfaces are used by Gauss’ Law,
which states that:which states that:
∫∫ ⋅=Φ=Φ AdEd
∫
Gauss’ Law
The flux of electric field through a closed
surface is proportional to the charge
enclosed.
Flux positive =>
out Flux negative
=> in
11. FluxFlux
Flux in Physics is used to two distinct ways.Flux in Physics is used to two distinct ways.
The first meaning is the rate of flow, such as the amount of water flowing inThe first meaning is the rate of flow, such as the amount of water flowing in
a river, i.e. volume per unit area per unit time. Or, for light, it is the amounta river, i.e. volume per unit area per unit time. Or, for light, it is the amount
of energy per unit area per unit time.of energy per unit area per unit time.
Let’s look at the case for light:Let’s look at the case for light:
12. September 26, 2007September 26, 2007
Flux of Electric FieldFlux of Electric Field
1. Which of the following figures correctly shows a positive1. Which of the following figures correctly shows a positive
electric flux out of a surface element?electric flux out of a surface element?
A.A. I.I.
B.B. II.II.
C.C. III.III.
D.D. IV.IV.
E.E. I and III.I and III.
θ
E
∆A
θ
E∆A
θ
E
∆A
θ
E
∆A
I. II.
III. IV.
13. September 26, 2007September 26, 2007
Mathematical Statement of Gauss’Mathematical Statement of Gauss’
LawLaw
The constant of proportionality in Gauss’ Law is ourThe constant of proportionality in Gauss’ Law is our
old friendold friend εε00..
Recall that I said that we would see later whyRecall that I said that we would see later why
Coulomb’s constant is written ?Coulomb’s constant is written ?
We can see it now by integrating the electric flux ofWe can see it now by integrating the electric flux of
a point charge over a spherical gaussian surface.a point charge over a spherical gaussian surface.
∫ =⋅
=Φ
enc
enc
qAdE
q
0
0
ε
ε
04
1
πε
=Ek
r
qenc
∫ ∫ ===⋅ encqrEdAEAdE 2
000 4πεεε
2
04
1
r
q
E enc
πε
=
dAEAdE =⋅
Solving for E gives Coulomb’s Law.
14. September 26, 2007September 26, 2007
Example of Gauss’ LawExample of Gauss’ Law
Consider a dipole with equal positive and negativeConsider a dipole with equal positive and negative
charges.charges.
Imagine four surfacesImagine four surfaces SS11,, SS22,, SS33,, SS44, as shown., as shown.
SS11 encloses the positive charge. Note that the fieldencloses the positive charge. Note that the field
is everywhere outward, so the flux is positive.is everywhere outward, so the flux is positive.
SS22 encloses the negative charge. Note that the fieldencloses the negative charge. Note that the field
is everywhere inward, so the flux through theis everywhere inward, so the flux through the
surface is negative.surface is negative.
SS33 encloses no charge. The flux through theencloses no charge. The flux through the
surface is negative at the upper part, and positivesurface is negative at the upper part, and positive
at the lower part, but these cancel, and there is noat the lower part, but these cancel, and there is no
net flux through the surface.net flux through the surface.
SS44 encloses both charges. Again there is no netencloses both charges. Again there is no net
charge enclosed, so there is equal flux going outcharge enclosed, so there is equal flux going out
and coming in—no net flux through the surface.and coming in—no net flux through the surface.
15. 1515
Coulomb’s LawCoulomb’s Law
Electric chargeElectric charge
• The positive chargeThe positive charge
and negative charge:and negative charge:
Matter is made of atoms.Matter is made of atoms.
Inside an atom, there is the nucleus that is surrounded by electrons.Inside an atom, there is the nucleus that is surrounded by electrons.
Inside the nucleus, there are two particles called proton and neutron. TheInside the nucleus, there are two particles called proton and neutron. The
smallest nucleus contains only one proton. This is the nucleus inside a hydrogensmallest nucleus contains only one proton. This is the nucleus inside a hydrogen
atom.atom.
Proton and electron attract each other. Proton and proton, electron and electronProton and electron attract each other. Proton and proton, electron and electron
repel each other. This is a property of these matters (proton and electron) and werepel each other. This is a property of these matters (proton and electron) and we
call itcall it chargecharge..
Electric charge is a property of matter that can cause attraction and repulsion.Electric charge is a property of matter that can cause attraction and repulsion.
We callWe call the charge carried by electrons “negative (-the charge carried by electrons “negative (-e)e) ”” andand the charge carried bythe charge carried by
protons “positive (+protons “positive (+ee)”.)”. Charge is a value, or a scalar, not a vector. It is fullyCharge is a value, or a scalar, not a vector. It is fully
described by a number.described by a number.
16. 1616
The unit of electric chargeThe unit of electric charge
The SIThe SI11
unit for charge is theunit for charge is the coulomb.coulomb. An electron or a proton has a charge ofAn electron or a proton has a charge of
magnitudemagnitude ee = 1.602 18×10= 1.602 18×10−19−19
C (coulombs). Some scientists, chemists inC (coulombs). Some scientists, chemists in
particular, use another unit, theparticular, use another unit, the esuesu oror electrostatic unitelectrostatic unit. One esu equals. One esu equals
3.335 64×103.335 64×10−10−10
C.C.
To provide you with an idea of the magnitude of a coulomb, approximatelyTo provide you with an idea of the magnitude of a coulomb, approximately
0.8 C of charge flows through a 100 watt light bulb every second. Or about 50.8 C of charge flows through a 100 watt light bulb every second. Or about 5
million trillion electrons every second. The rate of charges flowing through amillion trillion electrons every second. The rate of charges flowing through a
conductor is called a current. We will get to this a few chapters later.conductor is called a current. We will get to this a few chapters later.
No one has ever seen the charge, but we sure all see its effect in everydayNo one has ever seen the charge, but we sure all see its effect in everyday
life: electrostatic discharge in dry winter days to all the appliances (lights tolife: electrostatic discharge in dry winter days to all the appliances (lights to
motors to cell phones) that are powered by electricity.motors to cell phones) that are powered by electricity.
How much more do we know about the charge?How much more do we know about the charge?
17. 1717
Charge and charge upCharge and charge up
• When the numbers of electrons andWhen the numbers of electrons and
protons in an object are the same, weprotons in an object are the same, we
say that this object is (electrically)say that this object is (electrically)
neutral. When they are not, we call itneutral. When they are not, we call it
charged.charged.
• There are many ways to charge up anThere are many ways to charge up an
object. A demo here:object. A demo here:
In this demo, we rub away or rub inIn this demo, we rub away or rub in
electrons to make anelectrons to make an
object positively or negatively charged.object positively or negatively charged.
18. 1818
Conservation of electric chargesConservation of electric charges
• Electric charge is conserved.Electric charge is conserved. Charge can move between objects in the system,Charge can move between objects in the system,
but the net charge of the system remains unchanged. Charges cannot bebut the net charge of the system remains unchanged. Charges cannot be
created or destroyed in the system, because charge is just a property ofcreated or destroyed in the system, because charge is just a property of
electrons and protons. They both are matter and matter conserves. A remarkelectrons and protons. They both are matter and matter conserves. A remark
on anti-matter: matter and anti-matter annihilate into energy. So what’s moreon anti-matter: matter and anti-matter annihilate into energy. So what’s more
fundamental is the conservation of energy, but that’s beyond this class.fundamental is the conservation of energy, but that’s beyond this class.
8 = 10 + (-2), so O-21 decays to Ne-21 plus two electrons.
19. 1919
Movement of electric charges in matterMovement of electric charges in matter
• A conductor:A conductor: An object or material in which charge can flow relatively freely.An object or material in which charge can flow relatively freely.
Example: metal, carbon, …Example: metal, carbon, …
• An insulator:An insulator: An object or material in which charge does not flow freely. Example:An object or material in which charge does not flow freely. Example:
plastic, glass, …plastic, glass, …
• To ground:To ground: Charge flows from a charged object to the ground, leaving the objectCharge flows from a charged object to the ground, leaving the object
neutral.neutral.
• The ground:The ground: A neutral object that can accept or supply an essentially unlimitedA neutral object that can accept or supply an essentially unlimited
number of charges. The Earth functions as an electric ground. Application basednumber of charges. The Earth functions as an electric ground. Application based
on these physics concepts: the lightning rod on tall buildings.on these physics concepts: the lightning rod on tall buildings.
20. 2020
Electrostatics – forces between chargesElectrostatics – forces between charges
• Unlike charges attract; Like charges repel.Unlike charges attract; Like charges repel.
21. 2121
Forces between charges – charge inductionForces between charges – charge induction
• The force between charges provides a second way to charge up an object:The force between charges provides a second way to charge up an object: ThisThis
process is call induction.process is call induction.
• Charging an inductor:Charging an inductor:
• Charge rearrangement in insulators:Charge rearrangement in insulators:
22. 2222
To quantitatively study the forces between charges, we introduce the law ofTo quantitatively study the forces between charges, we introduce the law of
this chapter:this chapter:
The Coulomb’s Law of forces between two point charges in vector form:The Coulomb’s Law of forces between two point charges in vector form:
PLAY
ACTIVE FIGURE
2212
0
0
0
229
0
21
12
mNC1085428
space.freeoftypermittivitheis
4
1
CmN1098768
constant.Coulombthecalledis
tofromdirectedrunit vectotheis
.ofbecauseexperienceforcetheisHere
⋅×=
=⋅×=
−
/.
/.k
k
.qq
qq
e
ε
ε
πε
12
12
r
F
23. 2323
Discussions about Coulomb’s LawDiscussions about Coulomb’s Law
• The term point charge refers to a particle of zero size that carries an electricThe term point charge refers to a particle of zero size that carries an electric
chargecharge
• The force is inversely proportional to the square of the separation r between theThe force is inversely proportional to the square of the separation r between the
charges and directed along the line joining themcharges and directed along the line joining them
• The force is proportional to the product of the charges,The force is proportional to the product of the charges, qq11 andand qq22, on the two, on the two
particlesparticles
• The force is attractive if the charges are of opposite signThe force is attractive if the charges are of opposite sign
• The force is repulsive if the charges are of like signThe force is repulsive if the charges are of like sign
• Electrical forces obey Newton’s Third LawElectrical forces obey Newton’s Third Law
The force onThe force on qq11 is equal in magnitude and opposite in direction to the force onis equal in magnitude and opposite in direction to the force on qq22
• With like signs for the charges,With like signs for the charges,
the productthe product qq11qq22 is positive andis positive and
the force is repulsivethe force is repulsive
• With unlike signs for the charges,With unlike signs for the charges,
the productthe product qq11qq22 is negative andis negative and
the force is attractivethe force is attractive
2112 FF
−=
24. 2424
The Superposition PrincipleThe Superposition Principle
• The resultant force on any one charge equals the vector sum of theThe resultant force on any one charge equals the vector sum of the
forces exerted by the other individual charges that are presentforces exerted by the other individual charges that are present
• If there are four charges fromIf there are four charges from qq11 toto qq44, the resultant force on, the resultant force on qq11 is theis the
vector sum of all the forces exerted on it by other charges:vector sum of all the forces exerted on it by other charges:
Remember to add forces as vectors:Remember to add forces as vectors:
Problem solving templateProblem solving template
1 21 31 41= + +F F F F
Template
Step 1, formulas or related concepts.
Step 2, known quantities.
Step 3, direct application of the formulas/concept or the
condition to form an equation.
Step 4, vector involved?
Step 5, unit in the final answer correct? Answered all were
asked?
25. 2525
Example 1Example 1
X axis
Step 1, formula
Step 2, known quantities:
xF
rF
2
21
x
2
21
:axis-XtheAlong
,LawsCoulomb'
r
qq
k
r
qq
k
e
e
=
=
C.10601electronanofchargetheandconstantThe
m3m14m,6m)5(1distances
536
19
2312
321
−
×=
=−==−−=
−===
.ek
rr
,eq,eq,eq
e
26. 2626
Step 3, direct application of the formula twice:
Step 4, vector involved? Yes, and the answer is
given in vector form.
Step 5, unit? Answered all asked? Yes. SI is used.
Unit Newton for force.
x
xFFF
xxF
xxF
(N)10005
(N))10601)(
3
35
6
36
(10618
is2chargeonforcenetThe
(N)
3
35
10618
(N)
6
36
10618
28
219
22
11
32122
2
11
2
23
23
32
2
11
2
12
21
12
−
−−
−
−
×=
×
×−
+
×
×=+=
×−
×==
×
×==
.
..
ee
.
r
qq
k
ee
.
r
qq
k
e
e
27. 2727
Example 2Example 2
Particle 1 and 2 carry the
same amount of charge.
What is the value of q for the
particles to balance under
the electrical and
gravitational forces?
The gravitational constant
G = 6.67×10−11
N·m2
/kg2
Step 1, formulas:
Step 2, known quantities:
2
21
2
21
2
21
2
21
FandF:onlymagnitude
LawsNewton',LawsCoulomb'
r
mm
G
r
qq
k
r
mm
G
r
qq
k
e
e
==
== rFrF
ge
ge
Gk.r.mm e andConstantsm,001kg,00121 ===
