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AMIRAJ COLLEGE OF AMIRAJ COLLEGE OF ENG & TECH ENG & TECH
NAME: Alpesh PatelNAME: Alpesh Patel Er.No:151080106008Er.No:151080106008 Subject : EEE(2110005) Subject : EEE(2110005) Topic: Faraday’s, Coulomb’s, Gauss LawsTopic: Faraday’s, Coulomb’s, Gauss Laws
Guiden by: Anukar sirGuiden by: Anukar sir
Faraday’s LawFaraday’s Law
Area ChangeArea Change The sliding bar creates an emf The sliding bar creates an emf
by changing the area in the by changing the area in the magnetic field.magnetic field.• Constant magnetic fieldConstant magnetic field
The potential was due to the The potential was due to the time rate of change of area.time rate of change of area.
tABvBLV
Field ChangeField Change An emf can also be generated An emf can also be generated
by changing the magnetic field.by changing the magnetic field.
The time rate of change of the The time rate of change of the field through a fixed loop field through a fixed loop provides the potential.provides the potential.
tBAV
Field OrientationField Orientation The emf depends on the The emf depends on the
change in field or the change change in field or the change in area.in area.• Area perpendicular to the fieldArea perpendicular to the field
This suggests that the product This suggests that the product of the field and area of the field and area perpendicular matters.perpendicular matters.
t
ABtBAV
cos
Magnetic FluxMagnetic Flux The product of the field and area perpendicular to the The product of the field and area perpendicular to the
field is the field 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.
cosABM
AB M
Faraday’s LawFaraday’s Law The flux can be used to get The flux can be used to get
the induced emf.the induced emf.• Sign indicates polaritySign indicates polarity
This is Faraday’s Law of This is Faraday’s Law of induction.induction.
For multiple turns the emf is For multiple turns the emf is multiplied.multiplied.• NN turns of wire turns of wire• NN is the flux linkage is the flux linkage
tM
tN M
Coil FluxCoil Flux A circular flat coil has 200 A circular flat coil has 200
turns of wire with a total turns of wire with a total resistance of 25 resistance of 25 and an and an enclosed area of 100 cmenclosed area of 100 cm22. .
There is a perpendicular There is a perpendicular magnetic field of 0.50 T that is magnetic field of 0.50 T that is turned off in 200 ms.turned off in 200 ms.
Find the current induced in the Find the current induced in the coil.coil.
This problem has three parts.This problem has three parts.
To get the current from the To get the current from the resistance the voltage is resistance the voltage is needed.needed.
To get the voltage the flux is To get the voltage the flux is needed.needed.• Flux linkage works, tooFlux linkage works, too
Find the flux first.Find the flux first.
Flux to CurrentFlux to Current The magnetic flux is The magnetic flux is = BA= BA..
• = (0.50 T)(100 cm= (0.50 T)(100 cm22))• = (0.50 T)(0.010 m= (0.50 T)(0.010 m22))• = 0.0050 T m= 0.0050 T m22
The change in flux is The change in flux is negative since it is turned negative since it is turned off.off.
The induced emf isThe induced emf is• EE = = NN //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 comes The 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 ))• II = 0.20 A = 0.20 A
next
September 26, 2007September 26, 2007
Gauss’ LawGauss’ Law We are going to be most interested in We are going to be most interested in closedclosed
surfaces, in which case the outward direction surfaces, in which case the outward direction becomes self-evident.becomes self-evident.
We can ask, what is the electric flux out of such a We can ask, what is the electric flux out of such a closed surface? Just integrate over the closed closed surface? Just integrate over the closed surface:surface:
The symbol has a little circle to indicate that The 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 a The 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:
AdEd
Gauss’ LawThe flux of electric field through a closed surface is proportional to the charge enclosed.
Flux positive => out Flux negative => in
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 in The 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 amount a 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:
September 26, 2007September 26, 2007
Flux of Electric FieldFlux of Electric Field1. Which of the following figures correctly shows a positive 1. 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.
E
A
EA
E
A
E
A
I. II.
III. IV.
September 26, 2007September 26, 2007
Mathematical Statement of Mathematical Statement of Gauss’ LawGauss’ Law
The constant of proportionality in Gauss’ Law is our The constant of proportionality in Gauss’ Law is our old friend old friend ..
Recall that I said that we would see later why Recall 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 of We 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.
enc
enc
qAdE
q
0
0
041
Ek
r
qenc
encqrEdAEAdE 2000 4
204
1r
qE enc
dAEAdE
Solving for E gives Coulomb’s Law.
September 26, 2007September 26, 2007
Example of Gauss’ LawExample of Gauss’ Law Consider a dipole with equal positive and negative Consider a dipole with equal positive and negative
charges.charges. Imagine four surfaces Imagine four surfaces SS11, , SS22, , SS33, , SS44, as shown., as shown. SS11 encloses the positive charge. Note that the field encloses 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 field encloses the negative charge. Note that the field
is everywhere inward, so the flux through the is everywhere inward, so the flux through the surface is negative.surface is negative.
SS33 encloses no charge. The flux through the encloses no charge. The flux through the surface is negative at the upper part, and positive at surface is negative at the upper part, and positive at the lower part, but these cancel, and there is no net the lower part, but these cancel, and there is no net flux through the surface.flux through the surface.
SS44 encloses both charges. Again there is no net encloses both charges. Again there is no net charge enclosed, so there is equal flux going out charge 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.
1515
Coulomb’s LawCoulomb’s Law Electric chargeElectric charge
• The positive charge The 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. The Inside the nucleus, there are two particles called proton and neutron. The smallest nucleus contains only one proton. This is the nucleus inside a hydrogen smallest 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 electron Proton 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 we repel each other. This is a property of these matters (proton and electron) and we call it call 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 call We call the charge carried by electrons “negative (-the charge carried by electrons “negative (-e)e) ” ” and and the charge carried by the charge carried by protons “positive (+protons “positive (+ee)”.)”. Charge is a value, or a scalar, not a vector. It is fully Charge is a value, or a scalar, not a vector. It is fully described by a number. described by a number.
1616
The unit of electric chargeThe unit of electric charge
The SIThe SI11 unit for charge is the unit for charge is the coulomb.coulomb. An electron or a proton has a charge of An electron or a proton has a charge of magnitude magnitude ee = 1.602 18×10 = 1.602 18×10−19−19 C (coulombs). Some scientists, chemists in C (coulombs). Some scientists, chemists in particular, use another unit, the particular, use another unit, the esuesu or or 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, approximately To 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 5 0.8 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 a million 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 everyday No 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 to life: 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?
1717
Charge and charge upCharge and charge up
• When the numbers of electrons and When the numbers of electrons and protons in an object are the same, we protons 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 it neutral. When they are not, we call it charged. charged.
• There are many ways to charge up an There are many ways to charge up an object. A demo here:object. A demo here:
In this demo, we rub away or rub in In this demo, we rub away or rub in electrons to make anelectrons to make anobject positively or negatively charged.object positively or negatively charged.
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 be but the net charge of the system remains unchanged. Charges cannot be created or destroyed in the system, because charge is just a property of created or destroyed in the system, because charge is just a property of electrons and protons. They both are matter and matter conserves. A remark electrons and protons. They both are matter and matter conserves. A remark on anti-matter: matter and anti-matter annihilate into energy. So what’s more on 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.
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 object Charge 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 unlimited A neutral object that can accept or supply an essentially unlimited
number of charges. The Earth functions as an electric ground. Application based number 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.
2020
Electrostatics – forces between chargesElectrostatics – forces between charges• Unlike charges attract; Like charges repel.Unlike charges attract; Like charges repel.
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: This This
process is call induction.process is call induction.• Charging an inductor:Charging an inductor:
• Charge rearrangement in insulators:Charge rearrangement in insulators:
2222
To quantitatively study the forces between charges, we introduce the law To quantitatively study the forces between charges, we introduce the law of this chapter:of 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:
PLAYACTIVE FIGURE
22120
0
0
2290
21
12
mNC 1085428
space. free ofty permittivi theis 4
1CmN 1098768
constant. Coulomb thecalled is to from directedr unit vecto theis
. of because experience force theis Here
/.
/.k
k.qq
e
12
12
rF
2323
Discussions about Coulomb’s LawDiscussions about Coulomb’s Law• The term point charge refers to a particle of zero size that carries an electric The 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 the The 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 and and 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 on The force on qq11 is equal in magnitude and opposite in direction to the force on is 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 product the product qq11qq22 is positive and is positive and the force is repulsivethe force is repulsive
• With unlike signs for the charges, With unlike signs for the charges, the product the product qq11qq22 is negative and is negative and the force is attractivethe force is attractive
2112 FF
2424
The Superposition PrincipleThe Superposition Principle• The resultant force on any one charge equals the vector sum of the The 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 from If there are four charges from qq11 to to qq44, the resultant force on , the resultant force on qq11 is the is 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 template1 21 31 41 F F F F
TemplateStep 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?
2525
Example 1Example 1
X axis
Step 1, formula
Step 2, known quantities:x F
r F
221
x
221
:axis-X theAlong
, Law sCoulomb'
rqq
k
rqq
k
e
e
C. 10601electron an of charge theand constant The
m 3m 14 m, 6m )5(1 distances536
19
2312
321
.ek
rr,eq,eq,eq
e
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
xFF F
xx F
xx F
(N)10005
(N) )10601)(3
356
36(10618
is 2 chargeon forcenet The
(N) 3
3510618
(N) 6
3610618
28
21922
1132122
211
223
2332
211
212
2112
.
..
ee.rqq
k
ee.rqq
k
e
e
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:
221
221
221
221
F and F :only magnitude
Law sNewton' , Law sCoulomb'
rmm
Grqq
k
rmm
Grqq
k
e
e
r Fr F
ge
ge
Gk.r.mm e and Constants m, 001 kg, 00121
