Transcript Biot – Savart Law

PH0101 UNIT 2 LECTURE 2
 Biot Savart law
 Ampere’s circuital law
 Faradays laws of
Electromagnetic
induction
 Electromagnetic
waves,
 Divergence, Curl and
Gradient
 Maxwell’s Equations
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Biot – Savart Law
Biot - Savart law is used to calculate the magnetic field due
to a current carrying conductor.
According to this law, the magnitude of the magnetic field at
any point P due to a small current element I.dl ( I = current
through the element, dl = length of the element) is,
Y
Idl sin 
dB 
2
r
In vector
notation,
 0 Idl sin 
dB  .
4
r2
 0 idl  r
dB 
.
3
4
r
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I
B
I.dl C
A

r
dB
P
X
2
Ampere’s circuital law
 It states that the line integral of the magnetic field
(vector B) around any closed path or circuit is
equal to μ0 (permeability of free space) times the
total current (I) flowing through the closed circuit.
Mathematically,
 
 B. dl   0 I
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Faraday’s Law of electromagnetic induction
 Michael Faraday found that whenever there is a change in
magnetic flux linked with a circuit, an emf is induced
resulting a flow of current in the circuit.
 The magnitude of the induced emf is directly proportional
to the rate of change of magnetic flux.
 Lenz’s rule gives the direction of the induced emf which
states that the induced current produced in a circuit always
in such a direction that it opposes the change or the cause
that produces it.
d
induced emf (e)  
dt
d is the change magnetic flux linked with a circuit
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Electromagnetic waves
 According to Maxwell’s modification of Ampere’s
law, a changing electric field gives rise to a
magnetic field.
 It leads to the generation of electromagnetic
disturbance comprising of time varying electric and
magnetic fields.
 These disturbances can be propagated through
space even in the absence of any material
medium.
 These disturbances have the properties of a wave
and are called electromagnetic waves.
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Representation of Electromagnetic waves
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Description of EM Wave
 The variations of electric intensity and magnetic intensity
are transverse in nature.
 The variations of E and H are perpendicular to each other
and also to the directions of wave propagation.
 The wave patterns of E and H for a traveling
electromagnetic wave obey Maxwell’s equations.
 EM waves cover a wide range of frequencies and they
travel with the same velocity as that of light i.e. 3  10 8 m
s – 1.
 The EM waves include radio frequency waves,
microwaves, infrared waves, visible light, ultraviolet rays,
X – rays and gamma rays.
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Divergence, Curl and Gradient Operations
(i) Divergence
 The divergence of a vector V written as div V
represents the
V y

V
V z
x
scalar
quantity.
div V =   V =


x
y
P
z
P
P
(a) positive
d ivergence
(b) n egative
d ivergence
(c) zero divergence
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Example for Divergence
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Physical significance of divergence
 Physically the divergence of a vector quantity represents
the rate of change of the field strength in the direction of
the field.
 If the divergence of the vector field is positive at a point
then something is diverging from a small volume
surrounding with the point as a source.
 If it negative, then something is converging into the small
volume surrounding that point is acting as sink.
 if the divergence at a point is zero then the rate at which
something entering a small volume surrounding that point
is equal to the rate at which it is leaving that volume.
 The vector field whose divergence is zero is called
solenoidal
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Curl of a Vector field

i


Curl V =   V 
x
Vx

j

y
Vy

k

z
Vz
Physically, the curl of a vector field represents the rate of
change of the field strength in a direction at right angles to
the field and is a measure of rotation of something in a
small volume surrounding a particular point.
For streamline motions and conservative fields, the curl is
zero while it is maximum near the whirlpools
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Representation of Curl
Curl
(I) No rotation of the
(II) Rotation of the
paddle wheel represents
zero curl
paddle wheel showing
the existence of curl
(III) direction of curl
For vector fields whose curl is zero there is no rotation of
the paddle wheel when it is placed in the field, Such fields
are called irrotational
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Gradient Operator
 The Gradient of a scalar function  is a vector
whose Cartesian components are,
 
,
x y

and
z
Then grad φ is given by,
    
Grad    i
 j
k
x
y
z

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The magnitude of this vector gives the maximum
rate of change of the scalar field and directed
towards the maximum change occurs.
The electric field intensity at any point is given by,
E =  grad V = negative gradient of potential
The negative sign implies that the direction of E
opposite to the direction in which V increases.
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Important Vector notations in electromagnetism
1.
2.
 (   E )   (   E )   E
2
div grad S =

3.
4.
2S




 ( S V )  S(  V )  V (  S )
curl grad  = 0
  ( )  0
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Theorems in vector fields
Gauss Divergence Theorem

It relates the volume integral of the divergence of a vector
V to the surface integral of the vector itself.
 According to this theorem, if a closed S bounds a volume
, then
div V) d
= V  ds
(or)
(


V
)
d


V

ds



S
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Stoke’s Theorem
 It relates the surface integral of the curl of a
vector to the line integral of the vector itself.
 According to this theorem, for a closed path
C bounds a surface S,
s
(curl V)  ds =
C
V dl
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Maxwell’s Equations
Maxwell’s equations combine the fundamental
laws of electricity and magnetism .
The are profound importance in the analysis of
most electromagnetic wave problems.
These equations are the mathematical
abstractions of certain experimentally observed facts
and find their application to all sorts of problem in
electromagnetism.
Maxwell’s equations are derived from Ampere’s
law, Faraday’s law and Gauss law.
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Maxwell’s Equations Summary
Maxwell’s
Equations
1.
Equation from
electrostatics
2. Equation from
magnetostatics
3. Equation from
Faradays law
4. Equation from
Ampere
circuital law
Differential form
.D  
Integral form
 D.ds    dv
s
.B  0
v
 B.ds
0
s
B
 E 
t
D
 H  E
t
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B
 E . dl    t .ds
s
D
 H .dl   (  E  t ).ds
l
s
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Maxwell’s equation: Derivation
Maxwell’s First Equation
 If the charge is distributed over a volume V.
Let  be the volume density of the charge,
then the charge q is given by,
q=
  dv
v
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The integral form of Gauss law is,
11
φφEE  ds
ds
ρdv
ρdv


ss
εε00 v v
(1)
By using divergence theorem

s
E  ds 
 (   E ) dv
(2)
v
From equations (1) and (2),
1
 (   E )dv  ε  ρdv
0 v
v
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(3)
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
 E


(4)
(4)

0
div
E 

0

div E 
0
(5)
(5)
Electric displacement vector is
D  0 E
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(6)
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Eqn(5) ×
0 

 

div
E

 0 0div E 
  00
00
(or) div ( 0 E )  
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
((
D
D ))
(7)
This is the differential form of Maxwell’s I Equation.
From Gauss law
in integral form
 ds  ρdv
ρdv
sε0 Esε0 E ds
s



D
 ds
s ds
D

vv


dv
 
dv

v
v
This is the integral form of Maxwell’s I Equation
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Maxwell’s Second Equation
From Biot -Savart law of electromagnetism, the
magnetic induction at any point due to a current
element,
 o idl sin

dB =
4
r2
(1)

^




0
0
In vector notation,dB 
( idl  r )
(
idl

r
)
=
2
4 r
4 r 3

Therefore, the total induction B =
^
 0i
1
( 2 .dl  r )

4
r
(2)
This is Biot – Savart law.
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i
J 
A
(3)
replacing the current i by the current density J, the
current per unit area is
^
0 1
B
.(J  r ).dv [ i =J . A and I . dl = J(A . dl)

2
4 r

= J .dv]
Taking divergence on both
sides,
^
0
1
B 
 (
.J  r )dv

4 v
r2
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(4)
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 J 0
For constant current density
(5)

  B  0
Differential form of Maxwell’s’ second equation
By Gauss divergence theorem,

v

(  B ) dv   B.ds  0
(6)
s
Integral form of Maxwell’s’ second equation.
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Maxwell’s Third Equation
By Faradays’ law of electromagnetic induction,
e 
d
dt
(1)
By considering work done on a charge, moving
through a distance dl.
W
=

E .dl
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(2)
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If the work is done along a closed path,
emf =
 E .dl
(3)
The magnetic flux linked with closed area S due to the
Induction B =    B. ds
s
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(4)
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d
d
emf  e  

dt ddt
 B .ds
s

d
emf  e  


B .ds

s
dt
d B dt
= 
 ds
s dt


dB
E .dl   d B
.ds
s dt
E .dl  
.ds

l
 dt
(5)
(6)
s
(Integral form of Maxwell’s third equation)
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
E .dl
 (   E ).ds
=
s
  (   E ).ds   
s
Hence,
(7)
s
(Using Stokes’ theorem )
dB
.ds
dt
B
(   E ) 
t
(8)
Maxwell’s’ third equation in differential form
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Maxwell’s Fourth Equation
By Amperes’ circuital law,
B
We know,  0 
H
 H .dl  i
 B.dl  
0
i
(1)
l
(or) B = μ0 H
Using (1) and (2)
(2)
(3)
l
We know i =

J .ds
(4)
s
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 H .dl   J . ds
l
s
Using (3) and (4)
 H .dl   J . ds
We
WeKnow
Knowthat
that

 H .dl   J . ds

s
s

D
JJ 
EE D

tt
D
D
. dl EE..ds
ds

..ds
H
. dl
ds
J H
D
tt
E
l 
ss
ss
t

(5)


(6)
(6)
(7)
(7)
(Maxwell’s fourth equation in integral form)
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Using Stokes theorem,
 H .dl   (  H ).ds
l
(8)
(8)
s
(Using
(7) and
(Using
(7) (8))
and (8))
D
.ds
t
s
 (   H ). ds    E . ds  
s
s
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(9)
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The above equation can also be written as



D
 (   H ). ds    E  t  . ds
s

 


D
 (   H ). ds    E  t  . ds
s


D
  H  curl H   E 
t

(10)
(11)
Differential form of Maxwell fourth equation
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