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
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
Pan
dey
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
Dr. D K
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dey
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 : ???
Dr. D K
Pan
dey
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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Pan
dey
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
Dr. D K
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2 June 2021 IAPT, RC-4 : Workshop 7
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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2 June 2021 IAPT, RC-4 : Workshop 8 2 June 2021 8
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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dey
2 June 2021 IAPT, RC-4 : Workshop 9
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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dey
2 June 2021 IAPT, RC-4 : Workshop 10
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
Dr. D K
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dey
2 June 2021 IAPT, RC-4 : Workshop 11 2 June 2021 11
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
Dr. D K
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dey
2 June 2021 IAPT, RC-4 : Workshop 12
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
Dr. D K
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2 June 2021 IAPT, RC-4 : Workshop 13
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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dey
2 June 2021 IAPT, RC-4 : Workshop 14
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
Dr. D K
Pan
dey
2 June 2021 IAPT, RC-4 : Workshop 15
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
Dr. D K
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dey
2 June 2021 IAPT, RC-4 : Workshop 16
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
Dr. D K
Pan
dey
2 June 2021 IAPT, RC-4 : Workshop 17
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
Dr. D K
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dey
2 June 2021 IAPT, RC-4 : Workshop 18 2 June 2021 18
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
Dr. D K
Pan
dey
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
Dr. D K
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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
Dr. D K
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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
Dr. D K
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dey
2 June 2021 IAPT, RC-4 : Workshop 22
;
;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
Dr. D K
Pan
dey
2 June 2021 IAPT, RC-4 : Workshop 23
2
2
002
dt
EdE
2
2
002
dt
HdH
EM wave in free space and isotropic mediums
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
Dr. D K
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dey
2 June 2021 IAPT, RC-4 : Workshop 24
EHK 0
HEK 0
0E.K
0H.K
H & KE
E & KH
EK
HK
EM wave in free space and isotropic mediums
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
Dr. D K
Pan
dey
2 June 2021 IAPT, RC-4 : Workshop 25
Electromagnetic wave Energy density of EM wave in Free space
EM wave in free space and isotropic mediums
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
Dr. D K
Pan
dey
2 June 2021 IAPT, RC-4 : Workshop 26
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
Dr. D K
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dey
2 June 2021 IAPT, RC-4 : Workshop 27
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
Dr. D K
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2 June 2021 IAPT, RC-4 : Workshop 28
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
Dr. D K
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dey
2 June 2021 IAPT, RC-4 : Workshop 29
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
Pan
dey
2 June 2021 IAPT, RC-4 : Workshop 30
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
Dr. D
K P
ande
y
2 June 2021 IAPT, RC-4 : Workshop 31
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
Dr. D K
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dey
2 June 2021 IAPT, RC-4 : Workshop 32
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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dey
2 June 2021 IAPT, RC-4 : Workshop 33
EM wave in Anisotropic medium
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dey
2 June 2021 IAPT, RC-4 : Workshop 34
EM wave in Anisotropic medium
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Pan
dey
2 June 2021 IAPT, RC-4 : Workshop 35
EM wave in Anisotropic medium
1
2
10
20
11
22
2
1
V
Vn
Dr. D K
Pan
dey
2 June 2021 IAPT, RC-4 : Workshop 36
EM wave in Anisotropic medium
Dr. D K
Pan
dey
2 June 2021 IAPT, RC-4 : Workshop 37
EM wave in Anisotropic medium
Dr. D K
Pan
dey
2 June 2021 IAPT, RC-4 : Workshop 38
EM wave in Anisotropic medium
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dey
2 June 2021 IAPT, RC-4 : Workshop 39
EM wave in Anisotropic medium
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2 June 2021 IAPT, RC-4 : Workshop 40
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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2 June 2021 IAPT, RC-4 : Workshop 41
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
Dr. D K
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dey
2 June 2021 IAPT, RC-4 : Workshop 42
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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dey
2 June 2021 IAPT, RC-4 : Workshop 43
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(
Dr. D K
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dey
2 June 2021 IAPT, RC-4 : Workshop 44
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
Dr. D K
Pan
dey
2 June 2021 IAPT, RC-4 : Workshop 45
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
Dr. D K
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2 June 2021 IAPT, RC-4 : Workshop 46
)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
Dr. D K
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2 June 2021 IAPT, RC-4 : Workshop 47
Case 1: when (n1/n2)<1 ; wave : rare to denser n=Sini / Sinr =(n2/n1)>1 and i > r
Fresnel Equation…………. contd for Reflection and Refraction of EM wave
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
Dr. D K
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2 June 2021 IAPT, RC-4 : Workshop 48
Fresnel Equation…………. contd for Reflection and Refraction of EM wave
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
Dr. D K
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dey
2 June 2021 IAPT, RC-4 : Workshop 49
)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
Fresnel Equation…………. contd for Reflection and Refraction of EM wave
Dr. D K
Pan
dey
2 June 2021 IAPT, RC-4 : Workshop 50
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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dey
2 June 2021 IAPT, RC-4 : Workshop 51
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
Pan
dey
IAPT, RC-4 : Workshop
A Lot of Thanks for kind attention
2 June 2021 52