Ampere's Circuital Law states the relationship between the current and the magnetic field created by it. This law states that the integral of magnetic field density (B) along an imaginary closed path is equal to the product of current enclosed by the path and permeability of the medium.

1. Ampere’s Circuital Law
Electromagnetic Field Theory [EE-373]
Group No. 07
ALI HAMZA – (18143122-020)
ARBAB HASSAN – (18143122-035)
AHMED RAZA – (18813122-002)

2. Introduction
● Ampere’s Circuital Law states the relationship
between the current and the magnetic field created by
it.
● The law is named in honor of André-Marie Ampère,
who by 1825 had laid the foundation of
electromagnetic theory

3. Definition
“The integral around a closed path of the component of the magnetic field
tangent to the direction of the path is equals to µ times the current I intercepted by
the area within the path.”
Where;
● The integral ( ) is a line integral
● B.dl is a integrated around a closed loop called Amperian loop
● The current I is net current enclosed by the loop
0
.
B dl I




4. James Clerk Maxwell
● James Clerk Maxwell had derived that.
0
0
.
.
.
B dl I
B
dl I
H dl I



 
 



0
B
H

 

 
 

5. Ampere’s Law
● It alternatively says,
“The line integral of magnetic field intensity H about any closed
path is exactly equal to the direct current enclosed by that path.”
Mathematically,
.
H dl I



6. Introduction
Gauss’s Law
● It defines as:
“The electric flux passing through any closed surface is equal to the total
charge enclosed by that surface.”
.
S
S
D dS Q



7. Introduction
Gauss's Law Ampere’s Law
• The geometrical figure is a surface for Gauss’s law and a line for Ampere’s
Law.
• Gauss’s law is used to calculate the
electric field
• Gauss’s law can be used to derive the
electrostatic field from symmetric
charge distribution,
• Ampere’s law is used to calculate the
magnetic field.
• Ampere’s law can be used to derive
the magneto static field from
symmetric current distribution.

8. Applications
Here is a list of applications where you will find Ampere’s circuital law
being put into use.
 Current Carrying Conductor
 Thick Wire
 Solenoid
 Toroidal Solenoid

9. Infinite long Current Carrying Conductor
● Let us take an electrical conductor, carrying a current of I ampere. And take
an imaginary loop around the conductor. We also call this loop as Amperian
loop.

10. Continue…
● Then let consider the radius of the loop is and the flux density created at
any point on the loop due to current through the conductor is B.
● Consider an infinitesimal length dl of the Amperian loop at the same point.
 


11. Continue…
● At each point on the Amperian loop, the value of B is constant since the
perpendicular distance of that point from the axis of conductor is fixed, but the
direction will be along the tangent on the loop at that point.


12. Continue…
● The close integral of the magnetic field density B along the Amperian loop,
will be,
(Direction of B & dl is same at each point on the loop)
.
cos
B dl
Bdl 


cos0
(2 )
Bdl
B dl B 
 



13. Continue…
● Now, according to Ampere’s Circuital Law;
Therefore,
0
.
B dl I



0
0
2
2
2
B I
B I
I
H
 
 


 
 

14. N Current Carrying Conductors
● Instead of one current carrying conductor, there are N number of conductors
carrying same current I, enclosed by the path, then
2
NI
H


2
(2 )
2
I
H dl dl
I
I




  
 

15. Magnetic Field Intensity due to Coaxial
Transmission Line
● Let us consider the Cross section of a coaxial cable carrying a uniformly
distributed current I in the inner conductor and -I in the outer conductor. The
magnetic field at any point is most easily determined by applying Ampere’s
circuital law about a circular path.

16. Continue…
● In order to find the magnetic field of the conductor, we divide the
conductor as different cases.
Case 1: a
 
2
2
2
2
.
cos0
H dl I
H dl I
a
I
H dl
a




 





17. Continue…
2
2
2
(2 )
2
I
H
a
I
H
a







18. Continue…
Case 2:
.
cos0
2
H dl I
H dl I
H dl I
I
H


 





a b

 

19. Continue…
Case 3: b c

 
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
.
cos0
1
(2 )
2
H dl I
b
H dl I I
c b
b
H dl I
c b
c
H I
c b
I c
H
c b
 
 






 

    

 
 

 
 

 
 

  

 
 

  

 




20. Continue…
Case 4: ρ > c
.
0
H dl I I
H
 



21. Diagram
a
 
a b

 
b c

 
ρ > c

22. Magnetic Flux &
Magnetic Flux Density
AHMED RAZA
18813122-002

23. Magnetic flux & Magnetic flux density
● Magnetic flux (most often denoted as Φ). The number of
magnetic (flux) field lines which pass through a given cross-
sectional area A.
● The SI unit of magnetic flux is the weber (in derived units: volt-
seconds).
● The CGS unit is the Maxwell.

24. Magnetic flux & Magnetic flux density

25. Magnetic flux
● Consider an area ‘A’, placed in a
magnetic field.
● Let this area is divided into small
segments each of area DA.
● Flux through DA is the product of
area and the normal component of
field B, i.e

26. Continue…
A
Cos
B
A
B )
( 


 
A
B
OR
A
BCos
D

D
D

D
.

27. Continue…
● A = area of loop
● Φ= angle between B and the normal to the loop
● Now flux over whole area A is the sum of fluxes through
all elements ∆A i.e.
A
B
OR
A
B
A
B
A
B
D




D

D

D



D

D

D


.
......
.
.
.
......
3
3
2
2
1
1
3
2
1

28. Magnetic flux density
● It is defined as the amount of magnetic flux in an unit
area perpendicular to the direction of magnetic flow.
● The Magnetic Flux Density (B) is related to the Magnetic
Field (H) by:
● μ is the permeability of the medium (material) where we
are measuring the fields.
H
B 


29. Magnetic flux density
● The magnetic flux density is measured in Webers
per square meter [Wb/m^2], which is equivalent to
Teslas [T].
● The B field is a vector field, which means it has a
magnitude and direction at each point in space.
● The constant μ is not dimensionless and has the
defined value for free space, in henrys per meter
(H/m), of
m
H /
10
4 7


 


30. Solenoid & Toroid
ARBAB HASSAN
18143122-035

31. Magnetic field due to solenoid
● A solenoid consist of long conducting
wire made up of many loops packed
closely together. For coil that are
packed closely together magnetic field
is uniform and toward the center.
● Let us consider the a long straight
solenoid having ‘n’ turn per unit length
and carrying electric current ‘I’ as
shown in figure.

32. Magnetic field due to solenoid
● Direction of the field is given by the
right-hand rule. The solenoid is
commonly used to obtain a uniform
magnetic field

33. Amperian loop to determine the Magnetic
field
● Consider a rectangular path ABCD line
of induction such that
● AB=L= length of rectangular path
● the number of turn enclose by the
rectangle is nL. Hence the total electric
current following through the
rectangular path is nLI. According to
Ampere’s law

34. Amperian loop to determine the Magnetic
field
● According to ampere’s law

35. Amperian loop to determine the Magnetic
field
● Near the end of solenoid, the lines of field are closed
to each other. While the space where field lines are
far away from each other there magnetic field is
neglected.
● The path C-->D would be zero because no magnetic
field in this path.
● The path A--->D and B--->C magnetic field will also
be zero because the B and dl are perpendicular to
each other.

36. Continue…

37. A Toroid
● A toroid is a solenoid is bent into shape
of a hollow doughnut. Let us consider a
toroidal solenoid of average radius ‘r’
having center ‘O’ and carrying current
‘I’.
● Let us consider an amperian loop of
radius r and traverse in a clock wise
direction. Let N be the turn of the toroid.
Then total current following through
toroid ‘NI’.

38. A Toroid
● According to Ampere’s law
Inside the toroid

39. Outside the Toroid
● Magnetic field outside the toroid is
zero because there is no magnetic
field lines passing through outside the
toroid.
B=0