phy2009s buffler electromagnetism

8/3/2019 PHY2009S Buffler Electromagnetism

1/28

1

Part C:

A brief introduction to

Maxwells equations

PHY2009S

Fields and fluids

Andy BufflerDepartment of PhysicsUniversity of Cape Town


2/28

2

Just about the whole of electromagnetism is contained in the

Maxwell equations:

o

=E

i 0 =B

i

=t

BE

=o o o

t

+

EB j


3/28

3


4/28


5/28

5

Gauss law of electrostatics ... 2

By the divergence theorem: divS V

d dV= E a E i

1

div ( )oV V

dV dV = E r

( )div 0

oV

dV

=

rE

must be true for

any volume V

( )div

o

=

rE

Maxwell I

Electric field diverges outwards from positivecharge and inwards towards negative charge.


6/28

6

An animation of the motion of a positive charge moving past a massive chargewhich is also positive. The smaller charge is deflected away from the largercharge because of their mutual repulsion. This repulsion is primarily due to apressure transmitted by the electric fields surrounding the charges.


7/28

7

An animation of the motion of a negative charge moving past amassive positive charge. The negative charge is deflected toward thepositive charge because of the attraction between them. This attractionis primarily due to a tension transmitted by the electric fields

surrounding the charges.


8/28

8

Gauss law of magnetism

The net outward flux of any magnetic field across any closed

surface S is zero (there can be no magnetic monopoles).

0S

d =B a

i

Then by the divergence theorem:

Maxwell IIdiv 0=B

Magnetic fields never diverge or converge.

They always form closed loops (circulate).


9/28

9

The animation shows two co-axial wire loops carrying current in the samesense. The loops attract one another. We show the field configurationhere using the "iron filings" representation. The bottom wire loop carries

three times the current of the top wire loop.


10/28

10

The potential difference induced along a small section of a

circuit Cis given by .

Faradays law of electromagnetic induction

B

S

d = B a

iB

inducedt

=

where

An emf can induced in a closed circuit by changing magnetic

fields or by the motion of the circuit through a magnetic field.

The circuit referred to could be a closed electric circuit or

imaginary closed loop in space.

d

dE

i

inducedC S

d dt

= = E B a

i i

where the surface S is bounded by the circuit C.

It follows that induced electric fields are non-conservative.


11/28

11

Faradays law of electromagnetic induction ... 2

By Stokes theorem:

C S

d dt

= E B a

i i

( )C S

d d= E E a

i i

( ) =S S

d dt

E a B a

i i

= 0S

dt

+

BE a

i

must be true for

any circuit C

curl =t

BE

Maxwell III

Changing magnetic fields induce electric fields which curlaround the changing magnetic fields.


12/28

12

When the current is distributed continuously as a current

density , then

(The currentIacross any surface S bounded by the closed path Cis

the flux of across S.)

Amperes Law

Magnetic fields may be produced by electric currents:

The line integral of a magnetic field around any closed path

Cis equal to the permeability of free space multiplied by thecurrent I through the area enclosed by the path.

B

o

C

d I=B i

o

o

C S

d d= B j a

i i

( )j r

=

S

I dj a

i

( )j r

Then

It follows that magnetic fields produced by electric currents are

also non-conservative. Therefore the magnetic fields cannot be

derived by scalar potentials, but only vector potentials.


13/28

13

By Stokes theorem:

Amperes Law ... 2

o

C S

d d=

B j a

i i

( )

C S

d d= B B a

i i

( ) = oS S

d d B a j a

i i

must be true for

any circuit C

curl =o

B j Maxwell IV

(almost ...)

( )o = 0

S

d

B j a

i

Magnetic field lines curl around

electric currents (think of the magnetic

field around a conducting wire). I

B


14/28

14

Suppose we have five rings that carry a number of free positive charges that are not moving. Since there

is no current, there is no magnetic field. Now suppose a set of external agents come along (one for eachcharge) and simultaneously spin up the charges counterclockwise as seen from above, at the same timeand at the same rate, in a manner that has been pre-arranged. Once the charges on the rings start toaccelerate, there is a magnetic field in the space between the rings, mostly parallel to their common axis,which is stronger inside the rings than outside. This is the solenoid configuration.As the magnetic flux through the rings grows, Faraday's Law tells us that there is an electric field inducedby the time-changing magnetic field that is circulating clockwise as seen from above. The force on thecharges due to this electric field is thus opposite the direction the external agents are trying to spin therings up in (counterclockwise), and thus the agents have to do additional work to spin up the chargesbecause of their charge. This is the source of the energy that is appearing in the magnetic field betweenthe rings-the work done by the agents against the "back emf".

Over the time when the magnetic field is increasing in the animation, the agents moving the charges to a

higher speed against the induced electric field are continually doing work. The electromagnetic energythat they are creating at the place where they are doing work (the path along which the charges move)flows both inward and outward. The direction of the flow of this energy is shown by the animated texturepatterns. This is the electromagnetic energy flow that increases the strength of the magnetic field in thespace between the rings as each positive charge is accelerated to a higher and higher velocity.


15/28

15

The animation shows the magnetic field configuration around a

permanent magnet as it falls under gravity through aconducting non-magnet ring. The current in the ring is indicatedby the small moving spheres. In this case, the magnet is light,the ring has zero resistance, and the magnet levitates abovethe ring. The motions of the field lines are in the direction of the

local Poynting flux vector.


16/28

16

The animation shows the magnetic field configuration around aconducting non-magnetic ring as it falls under gravity in the magneticfield of a fixed permanent magnet. The current in the ring is indicated bythe small moving spheres. In this case, the ring is heavy and has zero

resistance, and falls past the magnet. The motions of the field lines arein the direction of the local Poynting flux vector.


17/28

17

The 4 Maxwell equations above express laws that came to

Maxwell from other sources. But Maxwells genius was to seethat law IV was incomplete.

Consider a capacitor being charged. As the charge flows to

the capacitor plates, a magnetic field rings the wire.But what about between the plates?

Does the magnetic field stop abruptly between the plates where

there is no current?

Maxwell though not ...

No magnetic field here?

I

B


18/28

18

Maxwell reasoned that if changing magnetic fields induce

electric fields (Faraday), then, symmetrically, changing

electric fields might induce magnetic fields.

Although there was no experimental evidence for this at the time,

Maxwell added an extra term to his fourth equation, saying that

magnetic fields also curl around changing electric fields:

This term generates a

magnetic field between thecapacitor plates as the

electric fields builds up.

Some years later, this magnetic field was detected ..!

curl =o o o

t

+

EB j

Maxwell IV


19/28

19

Maxwells equations:

I II

III IV

For time independent electric and magnetic fields:

I II

III IV

( )div

o

=

rE

div 0=B

curl =t

BE

curl =o o o

t +

EB j

( )div

o

=

rE

curl = 0E

div 0=B

curl =o

B j


20/28

20

The displacement current density term in Maxwells fourth

equation led to a very large unexpected physics jackpot ...

Imagine a single electric charge

being vibrated. In the space near the

vibrating charge, the charges

electric field is changing, so it

induces a magnetic field curling

around it.

But the magnetic field is also

changing, so it induces more

electric field, which induces moremagnetic field, etc, etc ...

The result is an electromagnetic wave of fields ripplingout from the vibrating charge at the speed of light.

o ot

E


21/28

21

Electromagnetic radiation continued ...

In empty space Maxwell III and IV become

=o o

t

EB

=t

BE

and

( ) ( )=t

E B

( ) = o ot t

EE

( )

2

2 2= o o t

EE E

i

22

2

=o o t

EE


22/28

22

This partial differential has electric field solutions that representwaves propagating in empty space with speed

These waves together with associated magnetic field waves

together describe constitute an electromagnetic wave.

2

22= o ot

EE

2

2

2=

o ot

BB

1

o o

c

= = 2.998 108 m s-1


23/28

23


24/28

24


25/28

25

showing that electrostatic field is a conservative field.

Since for any scalar field .

The electrostatic scalar potential

For time independent fields = 0 E

E

( ) = 0

When the electric charge distribution is described by a charge

density then we can use Gauss Law to obtain:

we can write

where the scalar field V is called the electrostatic potential.

= V E

o

=E

i

( ) oV =

i 2 o

V = Poissons

equationor

If = 0 everywhere, then 2 0V = Laplacesequation


26/28

26

where is the magnetic vector potential.

i.e. is not conservative.

The magnetostatic vector potential

The sources ofmagnetostatic fields are steady current densitydistributions:

Amperes law: = o B j

B

We also know that (no magnetic monopoles)0 =B

i

Since for any vector field ,( ) = 0 F

i F

we can write for some= B A

A

A


27/28


28/28

28

Electrostatic scalar potential Magnetostatic vector potential

Knowing and the

boundary conditions for V

Knowing and the

boundary conditions for Aj

2 o

V = 2 o = A j

A

V

B

= V E

E

= B A

phy2009s buffler electromagnetism

Documents