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P.Ravindran, PHY041: Electricity & Magnetism 1 March 2013: Maxwell equation related

Maxwell’s Equations

Gauss's law electricoS

qdε

E A

0 Gauss's law in magnetismS

S

d B A

Faraday's law

S

Bdddt

E s y

Ampere-Maxwell lawI E

dtdd μ ε μ

B s Ampere Maxwell lawIo o od μ ε μdt

B s

•The two Gauss’s laws are symmetrical, apart from the absence of the term for l ’ l fmagnetic monopoles in Gauss’s law for magnetism

•Faraday’s law and the Ampere‐Maxwell law are symmetrical in that the line integrals of E and B around a closed path are related to the rate of change of the


respective fluxes

• Gauss’s law (electrical):• The total electric flux through any closed

surface equals the net charge inside thatqd E Asurface equals the net charge inside that

surface divided by o• This relates an electric field to the charge

distribution that creates it

oS

dε

E A

• Gauss’s law (magnetism): • The total magnetic flux through any closedThe total magnetic flux through any closed

surface is zero• This says the number of field lines that enter

a closed volume must equal the number that 0d B Aqleave that volume

• This implies the magnetic field lines cannot begin or end at any point

S

• Isolated magnetic monopoles have not been observed in nature


• Faraday’s law of Induction:• This describes the creation of an electric

field by a changing magnetic fluxfield by a changing magnetic flux• The law states that the emf, which is the

line integral of the electric field around any closed path, equals the rate ofany closed path, equals the rate of change of the magnetic flux through any surface bounded by that path

• One consequence is the current induced Bdd

dt

E sin a conducting loop placed in a time‐varying B

dt

• The Ampere‐Maxwell law is a generalization of Ampere’s law

d• It describes the creation of a magnetic field by an electric field and electric currents

• The line integral of the magnetic field

I Eo o o

dd μ ε μdt

B s• The line integral of the magnetic field

around any closed path is the given sum


Maxwell’s Equation’s in integral form

A V

Q 1E dA dV

Gauss’s Law

o o

B dA 0

G ’ L f M iA

B dA 0 Gauss’s Law for Magnetism

BC A

d dE d B dAdt dt

Faraday’s Law

C Adt dt Ed dEB d I J dA

E

o encl o o o oC AB d I J dA

dt dt

A ’ LP.Ravindran, PHY041: Electricity & Magnetism 1 March 2013: Maxwell equation related

Ampere’s Law

Maxwell’s Equation’s in free space (no charge or current)( g )

E dA 0

Gauss’s LawA

E dA 0 Gauss s Law

AB dA 0

Gauss’s Law for Magnetism

Bd dE d B dA Faraday’s Law

C AE d B dA

dt dt Faraday s Law

Eo o o oC A

d dB d E dAdt dt

Ampere’s Law


Maxwell’s Equation’s In Differential Form

o

E

Gauss’s Law

B 0

Gauss’s Law for Magnetism

BE

F d ’ LEt

Faraday’s Law

o o oEB Jt

Ampere’s Law


o o o t

Reading Question

Electromagnetic induction was discovered byElectromagnetic induction was discovered by

1. Faraday.2 Henry2. Henry.3. Maxwell.4. Both Faraday and Henry.5 All h5. All three.


Reading QuestionReading QuestionElectromagnetic induction was discovered byElectromagnetic induction was discovered by

1. Faraday.2. Henry.3. Maxwell.4 Both Faraday and Henry4. Both Faraday and Henry.5. All three.


Reading QuestionReading QuestionThe direction that an induced current flowsThe direction that an induced current flows in a circuit is given by

1. Faraday’s law.. a aday s aw.2. Lenz’s law.3. Henry’s law.4. Hertz’s law.5. Maxwell’s law.


Reading QuestionReading Question

C rrents circ late in a piece of metal that isCurrents circulate in a piece of metal that is pulled through a magnetic field. What are these currents called?currents called?

1 Induced currents1. Induced currents2. Displacement currents3 Faraday’s currents3. Faraday s currents4. Eddy currents


Reading QuestionReading QuestionCurrents circulate in a piece of metal that is pulled through aCurrents circulate in a piece of metal that is pulled through a magnetic field. What are these currents called?

1. Induced currents2. Displacement currents3. Faraday’s currents4 Eddy currents4. Eddy currents5. This topic is not covered in Chapter 33.


Electromagnetic Induction

Consider a charge qprojected at velocity v into the magnetic field B shown •

•• • •• • •• • Bghere: •

•

•• • •• • •• •

•• • •• • •• •

Since the force is always di l t th

••

•• • •• • •• •

•• • •• • •• •perpendicular to the velocity, the charge will move in uniform circular motion:

•• • •• • •• •

•• • •• • •• •

••

motion:

F = mac

•• • •• • •• •

•• • •• • •• • ••

qvB = mv2

F = qvB sin

qvB = r

r = mvqB Particle in a


qMagnetic Field

Constant uniform Magnetic fieldConstant uniform Magnetic field

• B does not change the speed• It changes direction of motion• Thus particle should move in a circle if• Thus particle should move in a circle, if moves in a plane perpendicular to B.

vB2

||mvr

rmBvq || Bq

r||


Electromagnetic InductionInduced Electromagnetic Force (emf)

In fact, this very property is used in an instrument ll d thcalled the mass

spectrometer.

An unknown element is ionized, and accelerated by an applied voltage in the chamber S.

It is then projected into a known perpen‐dicular magnetic fieldmagnetic field.The radius is detected, and the mass calculated:

r = mvqB

m = rqBv


Angular frequencyg q y

Bqv x x x xx

x x x x xx

x x x x xx

x x x x xx

x

mBq

rv

x x x xxx x x x xxx

qBmr

x x x xxx

FBx x x xxxFB

mqBf

22

x x x xx

x x x x x

x

x

x x x xx

x x x x x

x

xm 22 x xx x


Application I‐Mass spectrometerpp p

To separate the ionized atomsR m


Application II - Cyclotron

mRBqmvK

221 222

2

E. O. Lawerence(1930)


Example 1Example 1

The semicircular conductor shown carries a current I. The closed circuit is exposed to a uniform magnetic field 0yBB. Determine (a) the magnetic force F1 on the straight section of the wire and (b) the force F2 on the curved section.

Solution to Example 1

• a) Using I BF a)

ˆ2ˆ

, Usingr

Im

xBF

N2ˆˆ2ˆ 001 IrBBIr zyxF

0ˆByB

N 22 001 IrBBIr zyxF

• b) :direction ˆ- in the is d ofproduct the zBl

N 2ˆsinˆ 00

00

2 IrBdrBIdI zzBlF


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