The document discusses James Clerk Maxwell's contributions to physics, particularly his formulation of Maxwell's equations, which describe the behavior of electric and magnetic waves. It also covers concepts such as displacement current, Poynting's theorem, and time-harmonic electric fields, along with the historical context of Maxwell's work. The conclusion emphasizes Maxwell's significant impact on modern physics despite the eventual discrediting of his theory of aether.

1. Maxwell’s contribution to
physics
Presented By:
Mayank panchal & Suhrid Mathur
Topics that will be covered:
About Maxwell
Maxwell’s Equations
Displacement current
Integral Form
Poynting Theorm
Time Harmonic Electric Field

2. James Clerk Maxwell
• Born in Edinburgh,
Scotland in 1831
• Attended Edinburgh
Academy, The
University of
Edinburgh, and
Cambridge University
• Published by the
young age of 14

3. More on Maxwell
• Maxwell differed from his contemporaries
in the nineteenth century
• Faraday & Ampere contributed to
Maxwell’s theories
• Much of his important work was
accomplished between the ages of 29 and
35

4. Maxwell’s equations
The behavior of electric and magnetic waves can be fully
described by a set of four equations (which we have learned
already).
B
E
t
∂
∇× = −
∂
D
H J
t
∂
∇× = +
∂
vD ρ∇ =g
0B∇ =gGauss’s Law for
magnetism
Gauss’s Law for
electricity
Ampere’s Law
Faraday’s Law of
induction

5. And the constitutive relations:
D Eε=
B Hµ=
J Eσ=
They relate the electromagnetic field to the properties of the material, in which
the field exists. Together with the Maxwell’s equations, the constitutive
relations completely describe the electromagnetic field.
Even the EM fields in a nonlinear media can be described through a
nonlinearity existing in the constitutive relations.

6. Conflict could be resolved by modifying Ampere’s
Law so that both electric current and displacement
current generate the magnetic field:
Displacement Current

7. For a time dependent electric field, a material medium
would become polarized, just as a dielectric does
- +
E
 For a constant E field, each pair of charges soon
equilibrates as shown above
 If the E field varies with time, then the charge
configurations are constantly in motion – displacement
current

8. L s
B
E dl ds
t∆ ∆
∂
= −
∂∫ ∫g gÑ
Gauss’s Law for
magnetism
Gauss’s Law for
electricity
Ampere’s Law
Faraday’s Law of
induction
Integral form
L s
D
H dl J ds
t∆ ∆
∂ 
= − + ÷
∂ 
∫ ∫g gÑ
v
S v
D ds dvρ
∆ ∆
=∫ ∫gÑ
0
S
B ds
∆
=∫ gÑ

9. It is frequently needed to determine the direction the power is
flowing. The Poynting’s Theorem is the tool for such tasks.
We consider an arbitrary
shaped volume:
B
E
t
∂
∇× = −
∂
D
H J
t
∂
∇× = +
∂
Recall:
We take the scalar product of E and subtract it from the scalar
product of H. B D
H E E H H E J
t t
∂ ∂ 
∇× − ∇× = − − + ÷
∂ ∂ 
g g g g
(6.9.1)
(6.9.2)
Poynting’s Theorem

10. Using the vector identity
( )A B B A A B∇ × = ∇× − ∇×g g g
Therefore:
( )
B D
E H H E E J
t t
∂ ∂
∇ × = − − −
∂ ∂
g g g g
( ) ( )2 21 1
2 2
B D
H E H H E E H E
t t t t
µ ε µ ε
∂ ∂ ∂ ∂
− − = − + = − +
∂ ∂ ∂ ∂
g g g g

11. ( )2 21
( )
2v v v
E H dv H E dv E Jdv
t
µ ε
∂
∇ × = − + −
∂∫ ∫ ∫V V V
g g
Application of divergence theorem and the Ohm’s law lead
to the PT:
( )2 2 21
( )
2s v v
E H ds H E dv E dv
t
µ ε σ
∂
× = − + −
∂∫ ∫ ∫V V V
gÑ
Here S E H= ×
is the Poynting vector – the power density
and the direction of the radiated EM fields
in W/m2
.
(6.11.3)

12. The Poynting’s Theorem states that the power that leaves a region is equal to
the temporal decay in the energy that is stored within the volume minus the
power that is dissipated as heat within it – energy conservation.
EM energy density is 2 21
2
w H Eµ ε = + 
Power loss density is 2
Lp Eσ=
The differential form of the Poynting’s
Theorem:
L
w
S p
t
∂
∇ + = −
∂
g

13. Time-harmonic EM fields
Frequently, a temporal variation of EM fields is harmonic; therefore, we
may use a phasor representation:
( , , , ) Re ( , , )
( , , , ) Re ( , , )
j t
j t
E x y z t E x y z e
H x y z t H x y z e
ω
ω
 =  
 =  
It may be a phase angle between the electric and the magnetic
fields incorporated into E(x,y,z) and H(x,y,z).
Maxwell’s Eqn in
phasor form:
( ) ( )E r j H rωµ∇× = −
( ) ( ) ( )H r j E r J rωε∇× = +
( ) ( )vE r rρ ε∇ =g
( ) 0B r∇ =g

14. Power is a real quantity and, keeping in mind
that: Re ( ) Re ( ) Re ( ) ( )j t j t j t j t
E r e H r e E r e H r eω ω ω ω
     × ≠ ×     
Since [ ]
*
Re
2
A A
A
+
=
complex conjugate
[ ] [ ]
* *
* * * *
( ) ( ) ( ) ( )
Re ( ) Re ( )
2 2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
4
E r E r H r H r
E r H r
E r H r E r H r E r H r E r H r
   + +
× = × ÷  ÷
   
× + × + × + ×
=
Taking the time average, we obtain the average
power as: *1
( ) Re ( ) ( )
2
avS r E r H r = × 

15. Therefore, the Poynting’s theorem in phasors is:
( ) ( )* 2 2 2
( ) ( )
s v v
E r H r ds j H E dv E dvω µ ε σ× = − − −∫ ∫ ∫V V V
gÑ
Total power radiated
from the volume
The power dissipated
within the volume
The energy stored
within the volume
Indicates that the power (energy) is reactive

16. Conclusion
• Maxwell provided us with modern physical
and mathematical equations
• Many contributions to physics even though
his belief in the existence of aether was not
valid
• Michelson-Morley experiment proved aether
wrong
• LIGO today uses similar apparatus and
encounters similar problems

17. Presented By:
Mayank panchal
Suhrid mathur