iapt, rc-4 : national revision workshop in physics

Electromagnetic Wave

IAPT, RC-4 : Workshop

Department of Physics, P. P. N. (PG) College,

Kanpur – 208 001, U.P., India Email: [email protected],

[email protected] Website: dkpandey.weebly.com

Dharmendra Kumar Pandey

2 June 2021 1

IAPT, RC-4 : NATIONAL REVISION WORKSHOP IN PHYSICS

mailto:[email protected]

2 June 2021 IAPT, RC-4 : Workshop

2

When the charge is static

Generated Field – Electric field

What happen ?

r̂r

q

4

1E

20

i2i

i

0

r̂r

q

4

1E

2

0 r

dq

4

1E

vd ; ds ; dldq

Method A Method B

Net outward electric flux through closed surface =q/0

0/qsd.E

dv1

sd.E0

dv1

dv E.0

0

E.

Method C

0E

E

0

2

r

d

4

1

0

dr

dE

0ld.E

Gauss law

Poisson’s equation

Point Charge

Many Point Charges

Charges distribution Dr. D K

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2 June 2021 IAPT, RC-4 : Workshop

3

When the charge is moving with constant speed.

Generated Field – Magnetic field

What happen ?

Method A Method B Method C

JB 0

AB

JA 02

r

dJ

4A 0

Ild.B 0

sd.Jld.B 0

sd.Jsd.B 0

Line integral of mag. Field over a closed loop = 0I

0B

r

lId

4A 0

0sd.B

0B

2

0

r

sin dl I

4dB

3

0

r

rld I

4B

Biot-Savart law

Gauss law

Ampere’s Law

Poisson’s equation

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When the charge moves with an acceleration then

IAPT, RC-4 : Workshop 2 June 2021 4

Field – Time varying electric and magnetic field

Mathematical expressions based on the law’s of

electrostatics, magnetostatics and electromagnetic

induction.

Useful in explanation of time varying electric and magnetic

fields or EM wave .

Maxwell Equation

What happen ?

Time varying electric and magnetic field : ??? Variation of E & D and B & H with time : ??? Differential equation for E & D and B & H : ??? Magnitude of E & D and B & H : ??? Direction of E & D and B & H : ???

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2 June 2021 IAPT, RC-4 : Workshop 5

First Maxwell Equation Gauss Law of electrostatics

sd.E sdEd

Net outward electric flux through closed surface =q/0

dvdq

dvqv

dv1

sd.E0

dv1

dv E.0

Integral form of Ist Maxwell equation

0

E.

D.

Differential form of Ist Maxwell equation ED ;ED 0

v

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2 June 2021 IAPT, RC-4 : Workshop 6 2 June 2021 6

Second Maxwell Equation Gauss Law of Magnetic field

sd.B sdBd

Monopole can not exists. Magnetic line of forces are closed curves. Net outward magnetic =0

0dv B.

Integral form of IInd Maxwell equation

Differential form of IInd Maxwell equation

HB ;HB 0

0sd.B

vv

0B.

0H. vv

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Third Maxwell Equation Faraday’s Law of EM Induction

Induced emf = - rate of change of magnetic flux

ld.Ee

sd.B

sd.Bdt

dld.E

sd.dt

Bdsd.E

dt

BdE

dt

de

v

Integral form of IIIrd Maxwell equation

Differential form of IIIrd Maxwell equation

dt

HdE

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Fourth Maxwell Equation Modified Ampere’s Law by Maxwell Line integral of magnetic field through closed path=0I

ld.Bfield magnetic of integral line

sd.JdI

sd.Jld.B 0

sd.Jsd.B 0

JB 0

This is Differential form of Ampere’s Law. Same result can be also obtained by Biot-Savart Law.

v Dr. D K

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J: Current density : Charge density

So, Ampere’s law and Biot savart Law are valid when charge density is constant or charges are moving with

constant/uniform.

Fourth Maxwell Equation ….contd. Modified Ampere’s Law by Maxwell …..contd

Equation of continuity : Based on conservation of charge

0dt

dJ.

J.B. 0

0J.

0dt

d

= constant

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Fourth Maxwell Equation ….contd. Modified Ampere’s Law by Maxwell …..contd

New current: Due change in electric flux with time New Current: Displacement Current

dt

dEA

dt

dEA

dt

dI 00

e0d

d0 JJB

dt

EdJ

dt

dE

A

IJ 0d0

dd

dt

Dd

dt

EdJ 0d

dJJH

Integral form and

Differential form of Fourth Maxwell equation

dtotal JJJ

sd.JJld.B d0

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Maxwell Equation

dJJB

dJJH

Integral form of Maxwell equation

sd.JJld.B d

Differential form of Maxwell equation

sd.Bdt

dld.E

dt

BdE

dt

HdE

dv1

sd.E

E.

D.

0B.

0H.

0sd.B

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E and B in terms of and A

dt

Ad

dt

)A(d

dt

BdE

AB

0dt

AdE

dt

AdE

dt

AdE

Gauge Transformation: And A are taken in such a way that E and B remains invariant.

Lorentz Gauge Transformation:

02

2

002

dt

d

0dt

dA. 00

Jdt

AdA 02

2

002

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1. Isotropic Medium:

Medium

Microscopic properties (a, ): direction independent Velocity of em wave : direction independent Example: air/free space/ vacuum, glass etc.

2. Anisotropic Medium Microscopic properties (a, ): direction dependent Example: quartz, calcite etc. Velocity of em wave : direction independent

3. Conducting Medium Attenuating medium for em wave Wave propagation vector: complex function E and H: damped oscillation Energy: decays with distance and time Example: conductors.

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Electromagnetic wave In Free space

Properties of free space

dt

EdJH

dt

HdE

E.

0H.

; ;1 ;1 0;J 0; ;0 00rr

Maxwell Equation

dt

EdH 0

dt

HdE 0

0E.

0H.

Maxwell Equation for free space

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Taking curl of IIIrd Maxwell Equation

2

2

002

00

0

dt

EdE

dt

Ed

dt

dE).(E.

Hdt

dE

2

2

002

dt

EdE

Electromagnetic wave In Free space: Differential Equation for E & H

Taking curl of IVth Maxwell Equation

2

2

002

00

0

dt

HdH

dt

Hd

dt

dH).(H.

Edt

dH

2

2

002

dt

HdH

Progresive wave Equation

2

2

2

2

dt

d

v

1

E and H are progresive wave t-r.Kj

0 e E)t,r(E

t-r.Kj0 e H)t,r(H

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Electromagnetic wave In Free space: Nature of E and H

dt

EdH 0

dt

HdE 0

0E.

0H.

EHK 0

HEK 0

0E.K

0H.K

H & KE

E & KH

EK

HK

lar.perpendicumutually are K and H ,E

n2

K

E

H K

t-zzk0x e EE

t-zzk0Y e HH

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0000 4

141v

C10310910v 897

Electromagnetic wave In Free space: velocity of em wave

2

2

002

dt

EdE

2

2

002

dt

HdH

2

2

2

2

dt

d

v

1 Comparing

C1

v00

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c1

v 00

Electromagnetic wave Impedance of Free space for em wave

medium space freeby offered ResistanceZ0

0

00

H

E

H

EZ

HEK 0

cvKH

E 000

cZ 00

120cZ

0

000

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2 June 2021 IAPT, RC-4 : Workshop 19 2 June 2021 19 2 June 2021 19

Electromagnetic wave Energy density of EM wave in Free space

eunit volumin stored Energyu

20m

20e H

2

1u & E

2

1u

v

me uuu

me00

002

0

20

m

e uu 1H

E

u

u

v

20eee Eu2uuu

v 2rms0

200

20 EE

2

1Eu

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2 June 2021 IAPT, RC-4 : Workshop 20 2 June 2021 20 2 June 2021 20 2 June 2021 20

Electromagnetic wave Poynting Vector in Free space

K along areaunit per gsincros PowerS

K along unit timeper areaunit per flowing EnergyS

v

HES

c

EnE

EKES

00

nZ

En

c

ES

0

2

0

2

nZ

En

c

ES

0

2rms

0

2rms

c density energy fluxenergy n c uS

v

K

E

H

S

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2 June 2021 IAPT, RC-4 : Workshop 21 2 June 2021 21 2 June 2021 21 2 June 2021 21 2 June 2021 21

Electromagnetic wave Poynting Theorem

K areaunit per gsincros PowerS

K along unit timeper areaunit per flowing EnergyS

S.t

uE.J

v

waveem into nsferedenergy tra of rateE.J

energy neticelectromag of change of ratet

u

surfaceboundary h theout throug flowingenergy S.

field/wave em into d transferePowerE.J

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;

;1 ;1

0;J 0; ;0

00

rr

;

;1 ;1

0;J 0; ;0

0r0r

rr

Properties of free space

Maxwell Equation

dt

EdH 0

dt

HdE 0

0E.

0H.

dt

EdH

dt

HdE

0E.

0H.

EM wave in free space and isotropic mediums

Free Space Isotropic medium

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2

2

002

dt

EdE

2

2

002

dt

HdH


Free Space Isotropic medium Wave Equation

2

22

dt

EdE

2

22

dt

HdH

C1

v00

CC1

vrr

Medium Impedance

120c

H

EZ

0

00

0

00

r

r00 Zv

H

EZ

Wave Velocity

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EHK 0

HEK 0

0E.K

0H.K

H & KE

E & KH

EK

HK


Free Space Isotropic medium Transverse nature of EM wave

EHK

HEK

0E.K

0H.K

H & KE

E & KH

EK

HK

Relation in E, B and K

HEK 0

BEK

c

En

v

EnEKB

HEK

BEK

v

EnEKB

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Electromagnetic wave Energy density of EM wave in Free space


Free Space Isotropic medium Energy density of EM wave

20m

20e H

2

1u & E

2

1u

20eme Eu2u & uu

2rms0

200 EE

2

1u

2m

2e H

2

1u & E

2

1u

2eme Eu2u & uu

2rms

20 EE

2

1u

Poynting Vector

HES

nZ

En

c

ES

0

2

0

2

nZ

En

c

ES

0

2rms

0

2rms

HES

nZ

En

v

ES

22

nZ

En

v

ES

2rms

2rms

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EM wave in conducting medium Medium Properties

; ;1 ;1 0;J 0; ;0 0r0rrr

Maxwell Equation

dt

EdEH

dt

HdE

/E.

0H.

Wave Equation

0dt

Ed

dt

EdE

2

22

0dt

Hd

dt

HdH

2

22

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Wave propagation vector

Solution of em wave Equation

)tr.(ir.0

)tr.K(i0 e e EeEE a

Propagation and attenuation constants

2/12 11(

2

a

aa

in)i(K

)tr.(ir.0

)tr.K(i0 e e HeHH a

2/12 11(

2

EM wave in conducting medium ..contd

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EM wave in conducting medium ..contd Penetration depth or skin depth: The distance at which amplitude of em wave becomes equal to 1/e times maximum value.

1/ eEeE 100

2 ; 1

a

For good Conductor

2

For Copper : m106.6

1MHzf mho/m; 108.5

5

7

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EM wave in conducting medium ..contd Refractive Index of conducting medium

)i(c

Kc

v

c*n a

Refractice Index

2n

iknc

ic

*n

a

c

n a

c

k

Attenuation constant

2 ; 1

a

For good Conductor

Dr. D K

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EM wave in conducting medium ..contd Impedence of conducting medium

4/12)1(

1Z

E and H are out of Phase tan

;

For Perfect Conductor

Phase difference between E and H

2 tan

2

1

2

1

42

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K P

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EM wave in conducting medium ..contd Energy density

t)-r.(cose E2

1u 2r.22

0e a

r.220e e E

4

1u

e2

m u1u

r.220

2m e E

4

11u

r.220

2 e E114

1u

r.220 e E

4

1u

For good Conductor

em u u

r.220 e E

22

1S

Avg. Poynting Vector

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EM wave in Anisotropic medium

1. EM wave properties depends on direction

2. 3. Permitivity is a tensor quantity.

0 0; ;0J

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1

2

10

20

11

22

2

1

V

Vn

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1. The normal component of electric displacement vector is not continuous at the interface but changes by an amount equal to the free surface charge density.

2. The normal component of magnetic induction B is continuous across the interface.

D-D 2n1n

0B-B 2n1n

Boundary condition for EM wave

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Boundary condition for EM wave Boundary condition for EM wave

3. The tangential component of electric field is continuous at the interface.

4. The tangential component of magnetic field strength is not continuous at the interface but changes by an amount equal to the component of the surface current density perpendicular to tangential component of H.

0E-E 2t1t

JH-H S2t1t

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Reflection and Refraction of EM wave For Incident wave

)tr.1K(i

011 eEE

1

111

EKB

11

111

EKH

For reflected wave )tr.1K(i

011 eEE

1

111

EKB

11

111

EKH

)tr.2K(i

022 eEE

2

222

EKB

11

111

EKH

For refracted wave Dr. D K

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Reflection and Refraction of EM wave

Since, tangential component of electric field is continuous at interface .

t2t1t1 )E()E()E(

)t2r.2k(i

t02

)t1r.1k(i

t1o

)t1r.1k(i

t01 e)E(e)E(e)E(

211

)r.2k(i

t02

)r.1k(i

t1o

)r.1k(i

t01 e)E(e)E(e)E(

t02t1ot01 )E()E()E(for ,Therefore

0z20z10z1 )r.k()r.k()r.k(

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Reflection and Refraction of EM wave

11 kk

r1i1 sinxksinxk

0z10z1 )r.k()r.k( ,If

Since wave propagation vector does not vary in same medium thus

ri sinsin

ri

0z20z1 )r.k()r.k( ,If

r2i1 sinxksinxk

1

2

r

i

k

k

sin

sin

2

1

1

2

r

i

v

v

k

k

sin

sin

11

22

2

1

r

i

v

v

sin

sinn

Law of reflection Law of refraction

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Fresnel Equation for Reflection and Refraction of EM wave

A: When E is perpendicular to plane of incidence

r

2

2i

1

1

r

2

2i

1

1

01

01

coscos

coscos

E

E

r

2

2i

1

1

i

1

1

01

02

coscos

cos2

E

E

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)sin(

sincos2

cosncosn

cosn2

E

E

ri

ri

r2i1

i1

01

02

Fresnel Equation…………. contd for Reflection and Refraction of EM wave

r

i

1

2

1

2021

sin

sin

n

n then if

)sin(

)sin(

cosncosn

cosncosn

E

E

ri

ri

r2i1

r2i1

01

01

For normal incidence ; i=r=0

21

21

01

01

nn

nn

E

E

21

1

01

02

nn

n2

E

E

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Case 1: when (n1/n2)<1 ; wave : rare to denser n=Sini / Sinr =(n2/n1)>1 and i > r


2differencePath ; Change Phase ve

E

E

01

01

Reflected and incident wave are in opposite phase. Refracted and incident wave are in same phase.

Case 2: when (n1/n2)>1 ; wave :denser to rare n=Sini / Sinr =(n2/n1)<1 and i < r

0 Change Phase veE

E

01

01

Reflected and incident wave are in same phase. Refracted and incident wave are in same phase.

0 Change Phase veE

E

01

02

0 Change Phase veE

E

01

02

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B: When E is parallel to plane of incidence

r

1

1i

2

2

r

1

1i

2

2

01

01

coscos

coscos

E

E

r

1

1i

2

2

i

1

1

01

02

coscos

cos2

E

E

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)cos()sin(

sincos2

cosncosn

cosn2

E

E

riri

ri

r1i2

i1

01

02

r

i

1

2

1

2021

sin

sin

n

n then if

)tan(

)tan(

cosncosn

cosncosn

E

E

ri

ri

r1i2

r1i2

01

01

For normal incidence ; i=r=0

12

12

01

01

nn

nn

E

E

12

1

01

02

nn

n2

E

E


Dr. D K

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Fresnel Equation…………. Contd…. for Reflection and Refraction of EM wave

Case 1:

Refracted and incident wave are in same phase.

0 Change Phase veE

E

01

02

Case 2: From both cases we found that -

)tan(

)tan(

E

Er

ri

ri

II01

01II

)sin(

)sin(

E

Er ;

ri

ri

01

01

1

21

riBprin

ntan

2 or

2 if

0r and 0rII Thus , if an un-polarized light incidents (rare to denser) at angle p=B then only electric vector perpendicular to plane of incidence will be reflected and light will become polarized. This angle is termed as angle of polarization or Brewster’s angle.

Dr. D K

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Fresnel Equation…………. Contd…. for Reflection and Refraction of EM wave

Case 3: if em wave travels from denser to rarer medium then it goes away from normal.

)tan(

)tan(

E

Er

ri

ri

II01

01II

)sin(

)sin(

E

Er ;

ri

ri

01

01

1

21

cirn

nsinor

2

1r and 1rII

1rR and 1rR22

IIII

Thus total energy is reflected at the interface of two mediums. This means, as the angle of incident wave increases, the intensity of reflected wave increases while intensity of refracted wave diminishes. The intensity of refracted wave becomes zero at i=c. But if i>c then wave completely goes in medium Ist, and follows law of reflection. This is TIR.

Dr. D K

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IAPT, RC-4 : Workshop

A Lot of Thanks for kind attention

2 June 2021 52

iapt, rc-4 : national revision workshop in physics

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