Maxwell’s equations

• Maxwell’s equations describe how electric and magnetic fields behave in the presence of charges and currentsand the relationship between electric and magnetic fields.

• They unify the description of electric and magnetic fields as originating from a common phenomenon.

• They constitute one of the milestones in the history of theoretical physics, along with Newton’s laws of motion,Einstein’s relativity theory, and quantum mechanics.

• They predict the existence of electromagnetic waves and provide a unified understanding of the origin of the variousforms of electromagnetic waves, from radio waves to visible light and gamma rays.

(Picture taken from Wikipedia: Electromagnetic spectrum - Wikipedia)

Electromagnetic spectrum

Maxwell’s 1st equation: Coulomb’s law

rer

qrE

2

04)(

charge +q

The electric field lines from a positive point chargeemanate radially away from the charge.

charge -q

rer

qrE

2

04)(

The electric field lines from a negative point chargeconverge radially towards the charge.

+Qre

r

qQrF

2

04)(

A positive charge +Qexperiences a radial forceaway from the charge q(equal charges repel each other)

The line direction tells us the direction of force on a positive chargeand the intensity (how dense the lines are) tells us the strength of the field.

+q -q

Electric field lines from a positive and a negative charge.

The electric field lines go out of the positive charge into the negative charge.

Derivation of Maxwell’s 1st equation

1) Place a point charge q at the center of a sphere of radius R:

R

2) Calculate the surface integral of the electric field linesthrough the surface of the sphere:

0

2

2

0

)(

2

0

)(

2

0)(

44

4

4)(

qR

R

q

dSR

q

eR

qedSrESd

VS

VS

rr

VS

independent of R

Maxwell’s 1st equation (integral form):

i

i

VS

qrESd

0)(

)(

S

True for any surfaceenclosing the charges

q

q

The equation also applies to this surface orany other surface enclosing the charge q.

S

Maxwell’s 1st equation (differential form):

)(00)(

)(11

)(SVi

i

VS

rdVqrESd

amount of chargein a little volume dV

charge density = charge per unit volume

Use Gauss formula: )(0)()(

)(1

)()(SVSVVS

rdVrEdVrESd

0

)()(

rrE

Maxwell’s 1st equation (differential form)

or also known as Gauss law

Example of the use of Maxwell’s 1st equation

R

R

Consider a sphere of radius R containing a uniform charge distribution of density ρ.We wish to figure out the electric field at a radial distance r from the center of the sphere when

r > R and r < R

r

The case when r > R:

S

rerErE

)()(

)(

3

)(3

4)(

SVSV

RdVrdVQ

0

2

)()()(

4)()(])([)(

Q

rrEdSrEerEedSrESdVS

r

VS

r

VS

Maxwell’s 1st:

2

04)(

r

QrE

It is the same as the electric field from a charge Q

located at the center of the sphere!

uniform charge density ρ

Total charge inside the red sphere:

The case when r < R:

R

r

S

rerErE

)()(

QR

rrdVrdVrq

SVSV

3

3

)(

3

)(3

4)()(

Total charge enclosed by the red sphere:

0

2

)()()(

)(4)()(])([)(

rqrrEdSrEerEedSrESd

VS

r

VS

r

VS

Maxwell’s 1st:

rR

Q

r

rqrE

3

0

2

0 44

)()(

It is linear in r !

R r

E(r)

r~2

1~

r

3

3

4RQ

Maxwell’s 2nd equation: No magnetic chargeMagnetic field lines of a magnetemanate from north to south

N

S

N

S

N

S

N

S

If a magnet is cut into two parts we get two magnets!The North and the South cannot be separated.

0)()()()(

SVVS

rBdVrBSd

no magnetic charge

0)( rB

Maxwell’s 3rd equation: Faraday’s law

A magnet moving towards a conducting loopinduces current in the loop in the direction shown.

dt

dV

V

I

Induced voltage V is equal tothe rate of change of magnetic fluxthrough the loop

)(

)(CS

rBSd The crucial point is that the magnetic flux

through the loop changes in time.If the flux is static, there is no current induced.

Faraday’s law:magnetic fluxthrough the loop C:

N S

B

If the magnet moves away from the loopthe induced current flows in the opposite direction.

C

When applying Faraday’s law, it is very importantto define the circuit.

(seen from the left side)

Maxwell’s 3rd equation: Faraday’s law

dt

dV

Induced voltage V is equal to the rate of changeof magnetic flux through the loop

)(

)(CS

rBSd

The crucial point is that the magnetic fluxthrough the loop changes in time.If the flux is static, there is no current induced.

Faraday’s law:

magnetic fluxthrough the loop C:

C

When applying Faraday’s law, it is very importantto define the circuit.

uniform magnetic fieldpointing towards us

I

I

V

Current in a conducting loop is induced when the magnetic flux through the loop is increased or decreased.

Derivation of Maxwell’s 3rd equation

)(

)(

CSt

rBSd

dt

dV

)(

)(

)(

CS

C

rESd

rErdV

A voltage in a circuit or loop C is the work donein bringing a positive unit charge around the loop.

)()(

)()(

CSCSt

rBSdrESd

from Stokes formula

t

rBrE

)()(

Maxwell’s 3rd (differential form)

Faraday’s law:

Lenz law: direction of induced current

uniform magnetic fieldpointing towards us

I

The magnetic field producedby the induced currenttries to keep the magnetic fluxthrough the loop constant,i.e., away from us to reducethe increasing flux.

uniform magnetic fieldpointing towards us

I

Here, the magnetic field producedby the induced current is towards usin order to increase the decreasing flux,to keep the flux constant.

Right-hand rule:

dt

dV

(Picture from Wikipedia:Right-hand rule - Wikipedia)

Biot-Savart law: the analogue of Coulomb’s law for current

2

0 ˆ

4)(

r

rldIrBd

ld

r

)(rBd

I

contribution to the magnetic field at rfrom a small current element lId

rer

dqrEd

2

04

1)(

compare with

Ampere’s law

I

r

IrB

2)( 0r

C

)()(2)(

rBrdrrBSC

This result can be derivedfrom Biot-Savart law

(see next slide)

IrBrdSC

0

)(

)(

The line integral of B around a loop C is given bythe current flowing through the surface enclosing the loop.

Note that the surface is arbitrary, as long as it encloses the loop C.

Ampere’s law

Ampere’s law is valid for a constant current but it breaks downwhen the current changes in time. The modification of Ampere’s law leads to Maxwell’s 4th equation.It is a very important contribution from Maxwell, which predicts the existence of electromagnetic waves.

)()()()( 0

)(

0

)(

rjrBrjSdrBSdCSCS

current densityfromStokes

I2

0 ˆ

4)(

r

rzdIrBd

R

z

2

2

2

sin1sin

Rrr

R

zd

r

dR

dzRz2sin

cot

2

0

00

2

sin2

4)(

R

I

Rd

IRB

Integrate from , which is equivalent to integrating from

0 toz

2 to0

(The factor of 2 arises because the contribution from is the same as the contribution from )

to0z0 toz

R

IRB

2)( 0 from Biot-Savart law

0

R

d

r

dz

2

2

0 sin

4)(

r

dzIRdB

R

dIRdB

sin

4)( 0

Fundamental problem with Ampere’s law

S1

S2

)(tI

)(tE

Switch onthe circuit

IrBrdSC

0

)(

)(

According to Ampere’s law,the line integral of the magnetic fieldaround the loop C is given by the currentflowing through the surface enclosing the loop:

C

There is no current flowing through surface S2.Ampere’s law is violated!

Notice that there is electric field piercing through S2

which decreases with time.

Current flows around the circuit which decays with timeas the charge in the capacitor is depleted.

chargedcapacitor

E

What is missing in Ampere’s law?

Consider the volume enclosed by the surface S = S1 + S2.

The current flowing out of the volume is given bydt

tdQtI

)()(

charge in the capacitor

)(

),()(VS

trjSdtI

)(

),()(SV

trdVdt

dtQ

dt

d

),(),()()(

trdt

ddVtrjSd

SVVS

=

),()(

trjdVSV

(Gauss) t

trtrj

),(),(

Continuity equation(conservation of charge)

)()( 0 rjrB

According to Ampere’s law

0)()]([ 0 rjrB

= 0 (mathematical identity)

Continuity equation (fundamental in physics)is not fulfilled!

Maxwell’s 4th equation

XjB

0

Modify Ampere’s law as follows:

such that the continuity equation is fulfilled:

0)( 0 XjB

tjX

From Maxwell’s 1st

t

E

tE

0

0

t

EX

0

t

EjB

00 Maxwell’s 4th

(Sometimes called“displacement current”)

Maxwell’s equations

00)(

QqESd

i

i

VS

0

E

0 B

t

BE

)()( CSSCt

BSdErd

t

EjB

00

t

EjSdBrd

CSSC

0)(

00

)(

0)(

VS

BSd

Coulomb’s law

No magnetic charge

Faraday’s law

Maxwell’scontribution

Differential form Integral formStokes-Gauss

E and B areinter-related

when theychange with

time

Electromagnetic wave equations in vacuum

In vacuum there is no charge and no current so that Maxwell’s equations become

0 E

0 B

t

BE

t

EB

00

Consider taking a rotation on Maxwell’s 3rd :

2

2

00)(t

EB

tE

EE

From Rule 2:

0 E

2

2

2

00

2

t

EE

Similarly, taking the rotation of Maxwell’s 4th yields 2

2

00

2

t

BB

Wave equations in 3D for E and B

Solutions to the wave equation

The wave equation in 1D has the general form2

2

22

2 1

t

f

cx

f

The solution is given by )()( ctxfctxff RL fL and fR are arbitrary functions.

moves to the rightmoves to the left with speed c

Check that it is a solution to the wave equation. Let

y

f

x

y

y

f

x

f LLL

ctxy

Chain ruleof differentiation:

2

2

2

2

y

f

x

y

y

f

yy

f

xx

f LLLL

y

fc

t

y

y

f

t

f LLL

2

22

2

2

y

fc

t

y

y

f

yc

y

f

tc

t

f LLLL

2

2

22

2 1

t

f

cx

f LL

(We have not specified fL soit is arbitrary)

(A second-order differential equationhas two independent solutions)

x=0

t=0

)()( xfctxf RR

t=1

x=c

)()( cxfctxf RR

In one second, the wave packet moves a distance c.In other words, its speed is c.

Let us plot fR (x-ct) for t=0 and t=1:

Consider now the electromagnetic wave equation in 1D for Ex

and assume that it depends only on x (in general Ex would depend on x, y, and z): 2

2

002

2

t

E

x

E xx

Comparison with the wave equation shows that: smcc /10311 8

0000

2

Maxwell not only predicted the existence of electromagnetic wavesbut also predicted the speed!

Harmonic solutions to the wave equation

)(exp tkxiAfR

)(exp tkxiAfL

2

2

22

2 1

t

f

cx

f

2

222

2

2 )(1

)(c

kiAc

ikA

ckk )( dispersion relation

Plane-wave solutions to the wave equation in 3D

)(exp)(exp),( tzkykxkiAtrkiAtrf zyx

r

any point on the plane has the same phase

phase

k

perpendicular to the plane

k

r

plane of equal phasemoves to the right

)(exp),( 0 trkiEtrE

)(exp),( 0 trkiBtrB

zyx EEEE 0000 ,,

zyx BBBB 0000 ,,

0 E

0 B

t

BE

t

EB

00

0Eki

0Bki

BiEki

EiBki

)(00

E and B are perpendicular to k

E is perpendicular to B

E

k

B

E

B

k

z

E

y

E

x

EE zyx

E

)(exp),( 0 tzkykxkiEtrE zyxxx

)(exp),( 0 trkiEtrE

zzyyxx EikEikEik 000

Eki

rk

)(exp),( 0 tzkykxkiEtrE zyxyy

)(exp),( 0 tzkykxkiEtrE zyxzz

Eit

E

E

y

E

x

Ee

x

E

z

Ee

z

E

y

Ee

EEE

zyx

eee

E xy

zzx

y

yzx

zyx

zyx

xyyxzzxxzyyzzyx EikEikeEikEikeEikEike

Eki

ki

For a plane wave (not in general):

it

|||| BEk

BiEki

magnitudes

BkE since k is perpendicular to E

cEEk

B

A typical value of E for a radio wave is 40 V/m B = 0.1 μT (very weak).Compare with the earth’s magnetic field of about 50 μT.A small bar magnet has about 0.01 T.

The direction of the electric field is used to define the direction of light polarization.

Magnitudes of E a μT nd B

Maxwell’s equations in a transparent material

E

0 B

t

BE

t

EjB

Still valid but the electric permittivity and magnetic permeability are modified depending on the material.

0000 , rr

Examples:Air 1.00059

Water 80.1

Pyrex (glass) 4.7

Silicon 11.7

r

The speed of light in a material is given by rr

mat

cc

1

Iron (Fe) 5000

Neodymium magnet 1.05

Aluminium 1.000 022

Nickel 100

r