Lecture physics a2 electromagnetic field and wave phd pham tan thi

Electromagnetic Field and WaveHo Chi Minh University of Technology... Maxwell’s EquationMaxwell discovered that the basic principles of electromagnetism can be expressed in terms of the

Electromagnetic Field and Wave

Ho Chi Minh University of Technology

Maxwell’s Equation

Maxwell discovered that the basic principles of electromagnetism can

be expressed in terms of the four equations that now we call Maxwell’s

(2) Gauss’s law for magnetic fields, showing no existence of magnetic

(4) Ampere’s law, including displacement current;

Maxwell’s Equations

Integral form:

I

~

E · d~ S = Qinside

"0

I

~

B · d~ S = 0

I

~

E · d~l = d B

dt

Differential form:

r · ~ E = ⇢

"0

r · ~ B = 0

r ⇥ ~ E = – @ ~ B

@t

r ⇥ ~ B = µ0J + µ ~ 0"0 @ ~ E

@t

Macroscopic Scale Microscopic Scale

I

~

B · d~l = µ0Ienclosed + µ0"0 d E

dt

Gauss’ Law

Gauss’ Law for Magnetism

Faraday’s Law

Ampere’s Law

Gauss’s Law for Electric Field

The flux of the electric field (the area integral of the electric field) over any closed surface (S) is equal to the net charge inside the surface (S) divided by the permittivity ε0

dS→

→

dx dy

d~ S = ˆ ndS = ˆ ndxdy

I

~

E · d~ S = Qinside

"0

~

E · dxdyˆn = Qinside"

0

~

E · dxdycos✓ = Qinside"

0

Edxdy = Qinside

"0

ES = E(4⇡r2) = Qinside

"0

E = Qinside

(4⇡r2)"0 Coulomb’s Law

Gauss’s law of magnetism states that the net magnetic flux through any closed surface is zero

Gauss’s Law of Magnetism

I

~

The number of magnetic field lines that exit equal to the

number for magnetic field lines that enter the closed

surface

I

~

E · d~ S = Qinside

"0

E→

Faraday’s Law

The electric field around a closed loop is equal to the negative of the rate of change of the magnetic flux through the area by the loop

W = F d = Eqd W

V = Ed = e.m.f

dt

I

~

E · d~l = d B

dt

dt

dt

~ Ed~l = Edlcos✓

= Edl

(θ = 0)

~

E · d~l

Ampère’s Law with Maxwell’s Correction

The line integral of magnetic field over a closed path is equal to the total current going through any surface bounded by the closed path

1 Time-changing electric fields induces magnetic fields

2 Displacement current

Conduction currents

Time changing electric fields also cause Magnetic field

=> Time changing electric fields is equivalent to a current We call it

dispalcement current

I

~

B · d~l = µ0Ienclosed + µ0"0 d E

dt

Ampère’s Law with Maxwell’s Correction

The line integral of magnetic field over a closed path is equal to the total current going through any surface bounded by the closed path

(when θ = 0)

I

~

B · d~l = µ0Ienclosed + d E

dt

Z Bdl = B(2⇡R) B(2⇡R) = µ0I

B = µ0I

2⇡R

I

~

B · d~l = µ0Ienclosed + µ0"0 d E

dt

~

B · d~l = Bdlcos✓ = Bdl

Ampère’s Law with Maxwell’s Correction

The line integral of magnetic field over a closed path is equal to the total current going through any surface bounded by the closed path

I

~

B · d~l = µ0"0 d E

dt + µ0Ienclosed

For S 1 :

I

~

B · d~l = µ0Ienclosed

For S 2:

I

~

B · d~l = 0

Two different situations in even one case!