[Electricity and Magnetism] Electrodynamics

MANMOHAN DASH, PHYSICIST, TEACHER !
Physics for ‘Engineers and Physicists’
“A concise course of important results”
Lecture - 2
Electrodynamics
Lectures given around 9.Nov.2009 + further
content developments this week; 21-
23.Aug.2015 !

Recapitulation
“We discussed the vectors and fields in the last
lecture”
Lecture - 1
Vector Calculus and Operations
I suggest ‘Lecture-1’ is followed first before
Lecture-2’ for a better understanding !
Take a quick look >> Vector Calculus Lecture-1 <<

Electric Field
In the last lecture presentation we discussed vector fields.
We saw how for every vector field there is a scalar field
comprehensively related to each other. Eg for every
potential energy field, which is a scalar field, there always
exists a vector field called a Force vector or force field.
Electric fields are the corresponding vector field of a
scalar field called electric potential or electric potential
field.
Electric fields are generated from various sources; static electric charge,
electric current, magnetic induction etc.

Electric fields from static electric charges
The electric fields that are produced from static electric
charges are determined by Coulomb’s Law of
electrostatics force. These are called electrostatics field.
Electric fields are generated from static electric charge, out of the 3 types
only +ve and –ve charges produce a Coulomb’s electrostatics force.
We have discussed recently that there are 3 kinds of
electric charges and not just two. Here (slide-2)
>> 3 types of electric charges <<
Accordingly only two types of charges give rise to
electrostatics force, the positive and the negative.

Electrostatic Fields in vacuum
The Coulomb’s
Force F is
proportional to
the two charges
(q, Q) and
inversely
proportional to
square of
separation r.
F is along line
joining 2 charges
The central force F is produced by charge Q on charge
q and electric field E is F/q. There are two scalar
fields energy E, potential V.
Accordingly there are two vector fields; F and E.

Coulomb’s Force Law, free space
We saw in last slide the Coulomb’s Force F is between 2 non-
zero charges which we denote as (1, 2), (q, Q) or (q1, q2).
F is a vector field, experienced by q, produced by Q. F is
related to scalar field E, the energy of configuration of q and
Q. Also F is deprecated to another vector field E by dividing
the charge (q) that experiences the force. E is related to its
own scalar field called the electric potential V.
So we have;
Its Q which produces electric force F and electric field E, on q.
r
r
Q
EEqFr
r
qQ
F ˆ
4
1
;;ˆ
4
1
2
0
2
0 



Coulomb’s Force; medium vs freespace !
We saw in last slide the Coulomb’s Force F is between 2 non-
zero charges in free space. Its slightly modified if the charges
in consideration are present in any media rather than
freespace.
We see that; we have two definitions of epsilon, the epsilon
with zero subscript, read as epsilon_nut, is the ‘permittivity of
free-space’ and the regular epsilon is the ‘permittivity of
medium’. The factor by which ‘permittivity of medium’ is
larger than ‘permittivity of free-space’ is called as ‘relative
permittivity’ and is denoted by epsilon_r. epsilon_r is also
called as ‘dielectric constant’ denoted by K.
r
r
Q
EEqFr
r
qQ
F ˆ
4
1
;;ˆ
4
1
22



100   Kr

Electric Displacement and Electric Flux!
Now we should define two more quantities in relation to the
electric field vector E.
Before that we need to state the values of the constants that
we discussed in last slide.
We note that the epsilon has particular values for particular
media. But epsilon_nut which specifies the permittivity of free
space has a very well specified value;
2
2
12
02
2
9
0
1085.8;100.9
4
1
Nm
C
C
Nm 
 


Electric Displacement and Electric Flux!
We already know E is the electric field vector. Here the charge
in the denominator is arbitrarily vanishing so as not to
intefere with the field F or E produced by source charge Q.
Similarly D is deprecated from E and called electric
displacement vector. It depends on the permittivity of the
space in which the E is being decided.
The electric flux Phi is defined as a surface integral (Lecture-1)
 



S
E
S
general
q
SdESdAEDED
q
F
E



 )(,,,lim 0
0

Electric Flux, Gauss Law of electrostatics!
Total electric
flux Phi over a
closed surface is
equal to 1/eps
times the net
charge enclosed
by the surface.
The surface is
called Gaussian
surface.
We evaluate the flux for a point charge Q from
the spherical symmetry of the electric field E and
find it to be;
22
0
2
4
1
;4
r
q
r
Eor
q
Er net
E

 
0
 net
S
E
q
SdE  


Gauss Law in differential form !
Gauss law in the last slide was in the integral form. We like
to cast it into a differential form. We also like to discuss its
form in the dielectric medium.
Gauss law can also be stated in terms of electric
Displacement Vector D (from above);
>> Total charge is from volume distribution of
charge. Now apply Gauss Divergence Theorem on the
surface integral in Gauss law for E. See GDT in (Lecture-1)
vacuummedium
net
S
E EDED
q
SdE

0
0
/, 

  
net
S
qSdD 

>> Gauss Law in Differential Form

V
net dVq 

Gauss Law in differential form !
By applying GDT we would have >>
So Gauss Law can be written as
Gauss Law in Differential Form <<
 
VS
dVESdE )(

0)(
1
)(
00
  VVV
dVEsodVdVE












 DEEE
mediumvacuum

0
00

Magnetic field.
A moving charge produces a magnetic
force. And in turn a moving charge
experiences magnetic force, produced by
any other moving charge. The stationary
charges do not interact magnetically.
Similar to electric field vector one can
define a magnetic field vector. We defined
electric field vector from electric force
vector. Hence the clue is to find the
magnetic force vector to define the
magnetic field vector.
Magnetic field is
also known as
magnetic induction
or magnetic flux
intensity. Its SI unit
is Tesla. Its lines of
force/flux are
shown below.

Magnetic field and magnetic force
A magnetic field B is defined in terms of the magnetic force
F which a moving test charge q experiences when it moves
in the field with velocity v. As we mentioned the SI unit of B
is Tesla.
Biot-Savart Law; The Biot-Savart law gives the elemental
magnetic field created, when an element of current induces
such a field. Like before Magnetic Permeability of free space
is denoted as; mu_nut. Mu is magnetic permeability in any
medium.
gauss
m
Wb
mA
N
T 4
2
101
.
11  BvqFB


A
Tm
r
rlId
Bd )104(,
4
7
03
0 


 





Magnetic intensity and magnetic flux
A magnetic field B is defined in terms of the magnetic force.
Like we defined an electric-displacement D vector from an
electric field vector E which we had defined from its electric
force vector F we can do the same for the magnetic field
vector B.
We define the magnetic intensity H from magnetic field
(We had defined B from magnetic force F) So F >> B >> H.
Like electric flux we define magnetic flux; its the surface
integral of the magnetic field B over the surface S. It has a
SI unit Weber (Wb). B is also called magnetic flux density.
met
Amp
unitSI
B
H ;,
0


  
S
B WbWeberunitSISdB .,



Magnetic flux and Gauss law of magnetism
There are no magnetic mono-poles or magnetic charges from
which a magnetic force emanates. The magnetic fields,
forces, flux emerge from electric charges in relative motion.
Charges in motion produce, electric forces, fields and flux as
well. Charges in rest produce electric fields, but in motion
produce electric & magnetic fields.  
S
SdB 0

0  VS
dVBSdB

The absence of magnetic mono-poles is represented by Gauss
Law of magnetism as stated above. Like we did for electric
flux, we convert the surface integral to a volume integral by
Gauss Divergence Theorem. This leads to the differential
form of Gauss law of magnetism. 0 B


We are in a pickle
Since there are no mag. mono-poles or magnetic charges
from which a magnetic force emanates we are in a pickle. As
we see, the Gauss law in magnetism does not correspond to
any kind of charge now. We can not then apply the same
treatment to magnetic field that we applied on electric fields?
Fortunately we have what it takes for us to be getting out of
the pickle. We have what we call Ampere’s Circuital Law
which would do the same for us that Gauss law in
electrostatics did. We could evaluate magnetic fields if we
have a symmetric situation; we have this amazing law.

Ampere’s Circuital Law
Line integral of a magnetic field B along a closed loop is
equal to mu_nut times the net electric current I enclosed by
the loop. We also note the same law for the H field we
defined some slides ago.
Consider an electric current flowing in a
straight line as-if emerging from the floor of the CCD where
you are drinking your coffee. Now Put your thumb (RH)
along it upwards and curl your palm. (so nails comes closer
to thumb) In the next slide we will see how the Ampere’s
circuital law we just discussed helps us determine the H, B.
net
C
net
C
IldHorIldB   ,0


Application

Magnetic field by Ampere’s Circuital Law
We have applied
here the Ampere’s
circuital law, by
recognizing the
circular symmetry
of the situation
from the straight
current. We are out
of the pickle now.
So the curl of the palm gives the direction of the magnetic
field as shown, ‘tangents’ to the circle. B is uniform across
the loop, the integral simply is; circumference of the loop.

Differential form of Ampere’s Circuital Law
To convert the integral form of Ampere’s law that we have
stated shortly, we have to use a similar trick as we used in
the case of electric field Gauss law. We convert the current I
into a surface integral, by recognizing the current density j is
the current per unit area. [j is a vector field, but I is not]
 
S
SdjI

ISdBldB
C
0
S
)(  

 
SC
SdjSdBldB

0
S
)( 
We apply the Stoke’s Theorem that we discussed in
(Lecture-1) to convert the line integral in Ampere’s circuital
law to the curl of the magnetic field B. This leads to the
differential form of the Ampere’s circuital law.
jB

0

Equation of Continuity !
Now that we see the flow of electric charge is what causes
magnetic forces (as well as electric forces) and these flow are
defined by current I and current density vector field j we
want to see the relation that establishes the familiar concept
fo conservation of electric charge.
 
S
SdjI

 
V
)( dVjSdj
S

 








VV
dV
t
dV
t

 )(
t
q-
I
In the second step we apply the Gauss Divergence Theorem
to convert the surface integral to a volume integral. In the 3rd
step we recognize the rate of flow of electric charge from
the volume V, through surface S, as the electric current.
0



t
j


Faraday’s Law of electromagnetic induction
Michael Faraday was an experimental Physicist who
stumbled upon the discovery that electric phenomena
induces magnetic effects as well as magnetic phenomena
induces the electric effects. Electricity and Magnetism were
considered separately until then. His work led to the
unification and this was together called as electromagnetism.
The basic observation that an emf (electromotive force, we
called electric potential field V, a scalar field, in Lect-1) is
developed when there is a variation in the magnetic flux is
known as Faraday’s law.
t
emf B




 )(

The emf is so induced that it opposes its own cause, much like
love developed in us opposes the very person who loves us.
The emf induced in a conducting loop is equal to the negative
of the rate of change of magnetic flux through the surface
enclosed by the loop. Its also called as Lenz’s law.
t
emf B




 )(

In slide 17 of (Lecture-1) we saw that electric field vector E is
the first derivative (in space) of the electric potential V. In
other words V is the path integral of the field vector E. [path
integral on slide 32-33 of (Lecture-1) ] We also know the
magnetic flux. This leads to an integral form of the Faraday’s
Law (of electromagnetic induction) and called as “integral
form of Maxwell-Faraday equation”. We use Stoke’s theorem
(like we did in case of Ampere’s Law) to reduce the integral
form to the differential form of Maxwell-Faraday Eq!







SC
S
B
C
SdB
t
ldEEqnsFaraday
wellMaSdBldE


'
xform;integral,
0;
)(Th;sStoke'







 
t
B
EeqnMFofformalDifferenti
SdB
t
SdEldE
SSC




Maxwell’s Displacement Current
Maxwell’s
displacement
current is a current
associated with a
time varying electric
field even if there is
no actual flow of
charges. Consider a
parallel plate
capacitor as shown.
The displacement
current I_d b/w
plates is found.
A current is produced between the
plates due to a variation in the electric
field E due to current I.

Modified Ampere’s Circuital Law
Now we have a reason why we must modify the Ampere’s
circuital law as we discussed it. This is because apart from an
actual current I (from motion of charge) there is a current
from the variation of E field as given above. This pseudo-
current also produces an actual magnetic field. So in effect the
variation in E produces the B field, as we mentioned at
beginning. Modified form is known as Ampere-Maxwell Law.
We saw that the displacement current flowing between the
plates is
This can also be written as a surface integral, for the general
case, by converting the area A for the case of capacitor into dS.
dt
dE
AId 0
t
IsoSdE
t
I E
d
S
d





 

 00 ;,


Modified Ampere’s Circuital Law
Instead of repeating the steps that we have followed before we
simply add the displacement current term at the right place.
Its easy;
We also realize that the term to the right of j is the
displacement current density. (the vector field)
Its also easy to see that by dividing mu on both sides of
differential form of M.A.C. law or Ampere-Maxwell law we get
the following differential law in terms of H.
)(;.
)(;
00
0
t
E
jBformDiff
IIldBformIntegral
C
d









t
E
jd





0
t
D
jH






Maxwell electromagnetic equations
When we summarize all the equations we discussed what we
get are called as Maxwell’s electromagnetic equations. These
are 4 in number but due to the fact that these can be stated in
terms of E or B field, D or H field and differential or integral
forms, they look like a lot. Lets summarize.
Differential Form [E, H, D, B appear. No Mu, eps appear]














j
t
D
H
t
B
E
Varyingtime





,0











0B
D
statesteady




Maxwell electromagnetic equations
Here are the differential and integral forms in terms of E and
B field only.
Differential and Integral Form [E, B only]
j
t
E
B
t
B
EBE















 0,0,










SCSC
SVS
Sd
t
E
jldBSdB
t
ldE
SdBdVSdE




)(,
0,
1
0
0




Thank you
We discussed extensively the electromagnetism course for an
engineering 1st year class. This is also useful for ‘hons’ and
‘pass’ Physics students.
This was a course I delivered to engineering first years, around
9th November 2009. I added all the diagrams and many
explanations only now; 21-23 Aug 2015.
Next; Lectures on ‘electromagnetic waves’ and ‘Oscillations
and Waves’. You can write me at g6pontiac@gmail.com or
visit my website at http://mdashf.org

[Electricity and Magnetism] Electrodynamics

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