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Chapter 34

- The Laws of Electromagnetism
- Maxwells Equations
- Displacement Current
- Electromagnetic Radiation

The Electromagnetic Spectrum

infra -red

ultra -violet

Radio waves

g-rays

m-wave

x-rays

The Equations of Electromagnetism (at this point

)

Gauss Law for Electrostatics

Gauss Law for Magnetism

Faradays Law of Induction

Amperes Law

The Equations of Electromagnetism

..monopole..

Gausss Laws

1

?

2

...theres no magnetic monopole....!!

The Equations of Electromagnetism

.. if you change a magnetic field you induce an

electric field.........

Faradays Law

3

Amperes Law

4

.......is the reverse true..?

...lets take a look at charge flowing into a

capacitor...

...when we derived Amperes Law we assumed

constant current...

...lets take a look at charge flowing into a

capacitor...

...when we derived Amperes Law we assumed

constant current...

E

B

.. if the loop encloses one plate of the

capacitor..there is a problem I 0

Side view (Surface is now like a bag)

Maxwell solved this problem by realizing that....

Inside the capacitor there must be an induced

magnetic field...

How?.

Maxwell solved this problem by realizing that....

Inside the capacitor there must be an induced

magnetic field...

How?. Inside the capacitor there is a changing E

?

B

A changing electric field induces a magnetic

field

E

Maxwell solved this problem by realizing that....

Inside the capacitor there must be an induced

magnetic field...

How?. Inside the capacitor there is a changing E

?

B

A changing electric field induces a magnetic

field

E

where Id is called the displacement

current

Maxwell solved this problem by realizing that....

Inside the capacitor there must be an induced

magnetic field...

How?. Inside the capacitor there is a changing E

?

B

A changing electric field induces a magnetic

field

E

where Id is called the displacement

current

Therefore, Maxwells revision of Amperes Law

becomes....

Derivation of Displacement Current

For a capacitor, and .

Now, the electric flux is given by EA, so

, where this current , not being associated with

charges, is called the Displacement current,

Id. Hence and

Derivation of Displacement Current

For a capacitor, and .

Now, the electric flux is given by EA, so

, where this current, not being associated with

charges, is called the Displacement Current,

Id. Hence and

Maxwells Equations of Electromagnetism

Gauss Law for Electrostatics

Gauss Law for Magnetism

Faradays Law of Induction

Amperes Law

Maxwells Equations of Electromagnetismin Vacuum

(no charges, no masses)

Consider these equations in a vacuum.....

......no mass, no charges. no currents.....

Maxwells Equations of Electromagnetismin Vacuum

(no charges, no masses)

Electromagnetic Waves

These two equations can be solved simultaneously.

The result is

Electromagnetic Waves

B

E

Electromagnetic Waves

B

E

Special case..PLANE WAVES...

satisfy the wave equation

Maxwells Solution

Plane Electromagnetic Waves

Ey

Bz

c

x

Static wave F(x) FP sin (kx ?) k 2?

? ? k wavenumber ? wavelength

Moving wave F(x, t) FP sin (kx - ?t ) ?

2? ? f ? angular frequency f frequency v

? / k

F

v

Moving wave F(x, t) FP sin (kx - ?t )

x

What happens at x 0 as a function of time?

F(0, t) FP sin (-?t)

For x 0 and t 0 ? F(0, 0) FP sin (0) For x

0 and t t ? F (0, t) FP sin (0 ?t) FP

sin ( ?t) This is equivalent to kx - ?t ? x

- (?/k) t F(x0) at time t is the same as

Fx-(?/k)t at time 0 The wave moves to the

right with speed ?/k

Plane Electromagnetic Waves

Ey

Bz

Notes Waves are in Phase, but fields

oriented at 900. k2p/l. Speed of wave is

cw/k ( fl) At all times EcB.

c

x