EMFT | Ampere's Circuital Law

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  

4. James Clerk Maxwell James Clerk Maxwell had derived that .

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,

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 .”

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 A mperian 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 . C onsider an infinitesimal length dl of the A mperian loop at the same point.

11. Continue … At each point on the A mperian 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 A mperian loop, will be , (Direction of B & dl is same at each point on the loop)

13. Continue… Now, according to Ampere’s Circuital Law; Therefore,

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

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:

17. Continue…

18. Continue … Case 2:

19. Continue… Case 3:

20. Continue… Case 4:  

21. Diagram  

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  D A. Flux through  D A is the product of area and the normal component of field B, i.e

26. Continue…

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.

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.

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

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