Magnetic field intensity, flux density, magnetic materials and Boundary conditions

1. R.M.K. COLLEGE OF ENGINEERING AND TECHNOLOGY
RSM Nagar, Puduvoyal - 601 206
DEPARTMENT OF ECE
EC 8451
ELECTRO MAGNETIC FIELDS
UNIT III
MAGNETOSTATICS
Dr. KANNAN K
AP/ECE

2. UNIT III
MAGNETOSTATICS
Lorentz Force equation
Law of no Magnetic Monopoles
Ampere's law, Vector Magnetic Potential
Biot-Savart law and its Applications
Magnetic Field intensity and idea of Relative Permeability
Magnetic Circuits and Behaviour of Magnetic materials
Boundary Conditions
Inductance and Inductors
Magnetic Energy
Magnetic Forces and Torques

3. Magnetic Field Intensity
Idea of Relative Permeability
Magnetic Circuits
Behaviour of Magnetic Materials
Boundary Conditions

4. Magnetic Field Intensity
Magnetic Field
The region around the magnet – Influence of the magnet
Represented by Magnetic field intensity and Magnetic flux density
Magnetic Lines of force
Magnetic field represented by imaginary lines
around the magnet
Magnetic flux lines
Lines of force around the magnet
Magnetic Field Intensity (H)
Force experienced by a unit north pole of one weber strength
Quantitative measure of strength or weakness of magnetic field
Ratio of the MMF needed to create a certain Flux Density (B) within a
particular material per unit length of that material.
Measured in Newtons/weber (N/Wb) or (A/M)

5. Magnetic Flux Density (B)
Total magnetic lines of force (ie) magnetic flux crossing a unit area
Measured in Weber/Sq.meter (Or) Tesla
Fundamental postulate of steady magnetic field specify (i) Divergence (ii) Curl of B in free space
∇. 𝐵 = 0 and ∇ 𝑋 𝐵 = 𝜇0J
𝜇0 = Permeability of free space
= 4𝜋 X 10-7 (H/M)
∇ 𝑋 𝐵 = 𝜇0J J = Current Density (A/M)
= 𝜇0(J + Jm)
1/ 𝜇0 (∇ 𝑋 𝐵) = J + Jm
= J + ∇ 𝑋 𝑀 ; 𝑀 = 𝑀𝑎𝑔𝑛𝑒𝑡𝑖𝑧𝑎𝑡𝑖𝑜𝑛
= Net dipole moment per unit volume
J = ∇ (B/ 𝜇0 - M)

6. Now we define new fundamental field quantity, the magnetic field intensity (H)
H = B/ 𝜇0 - M (A/M)
∴ ∇ 𝑋 𝐻 = J (A/m2 )
𝛻. 𝐵 = 0 and 𝛻 𝑋 𝐵 = J --- Magnetostatics
Taking surface integral on both sides
𝑆
∇ 𝑋 𝐻 . 𝑑𝑠 = 𝑆
𝐽. 𝑑𝑠 (OR)
According to Stokes theorem
𝐶
𝐻. 𝑑𝑙 = 𝐼 (𝐴)
---- C is contour bounding the Surface S ,I - total free current passing
through S
----- Ampere’s Circuital Law

7. Magnetic properties of the medium are linear and isotropic, then
M ∝ H
M = ℵ𝑚 𝐻
ℵ𝑚 = Magnetic Susceptibility = No dimension
∴ 𝐻 = B/ 𝜇0 − M
= B/ 𝜇0 - ℵ𝑚 𝐻
B/ 𝜇0 = H + ℵ𝑚 𝐻
B = 𝜇0H(1+ ℵ𝑚 ) = 𝜇0H 𝜇r ; 𝜇r = 1+ ℵ𝑚
B = 𝝁H ; 𝝁 = 𝝁𝟎 𝝁𝐫

9. MAGNETIC CIRCUITS
The closed path followed by magnetic lines of forces or magnetic flux is called magnetic circuit.
A magnetic circuit is made up of magnetic materials having high permeability such as iron, soft
steel, etc.
Magnetic circuits are used in various devices like electric motor, transformers, relays, generators
galvanometer, etc.
Analysis of Magnetic circuits based on the following equations are
∇. 𝐵 = 0 and ∇ 𝑋 𝐻 = J
Here ∇ 𝑋 𝐻 = J – Converts into Amperes circuital law.
If the closed path C is chosen to enclose N turns of winding
carrying current I that exits a magnetic circuit, then
𝐶
𝐻. 𝑑𝑙 = 𝐼𝑁 = 𝓇𝒎
𝓇𝒎 = Magneto Motive Force

10. BASIS MAGNETIC CIRCUIT ELECTRIC CIRCUIT
Definition
The closed path for magnetic flux is called
magnetic circuit.
The closed path for electric current is
called electric circuit.
Relation Between Flux and
Current
Flux = mmf/reluctance Current = emf/ resistance
Units Flux φ is measured in weber (wb)
Current I is measured in amperes
MMF and EMF
Magnetomotive force is the driving force and is
measured in Ampere turns (AT)
Mmf =ʃ H.dl
Electromotive force is the driving force
and measured in volts (V)
Emf = ʃ E.dl
Reluctance and Resistance
Reluctance opposes the flow of magnetic flux S
= l/aµ and measured in (AT/wb)
Resistance opposes the flow of current
R = ρ. l/a and measured in (Ώ)
Relation between Permeance
and Conduction
Permeance = 1/reluctance Conduction = 1/ resistance
Analogy Permeability
Conductivity
Analogy Reluctivity
Resistivity
Density Flux density B = φ/a (wb/m2)
Current density J = I/a (A/m2)
Intensity Magnetic intensity H = NI/l
Electric density E = V/d

11. Drops Mmf drop = φS Voltage drop = IR
Flux and Electrons
In magnetic circuit molecular poles are
aligned. The flux does not flow, but sets up in
the magnetic circuit.
In electric circuit electric current flows in the
form of electrons.
Examples
For magnetic flux, there is no perfect
insulator. It can set up even in the non
magnetic materials like air, rubber, glass etc.
For electric circuit there are a large number of
perfect insulators like glass, air, rubber, PVC
and synthetic resin which do not allow it to
flow through them.
Variation of Reluctance
and Resistance
The reluctance (S) of a magnetic circuit is not
constant rather it varies with the value of B.
The resistance (R) of an electric circuit is
almost constant as its value depends upon the
value of ρ. The value of ρ and R can change
slightly if the change in temperature takes
place
Energy in the circuit
Once the magnetic flux sets up in a magnetic
circuit, no energy is expanded. Only a small
amount of energy is required at the initial
stage to create flux in the circuit.
Energy is expanding continuously, as long as
the current flows through the electrical
circuit.
This energy is dissipated in the form of heat.
Applicable Laws Khirchhoff flux and mmf law is followed
Khirchhoff voltage and current law is
followed. (KVL and KCL)
Magnetic and Electric
lines
Magnetic lines of flux starts from North pole
and ends at South pole.
Electric lines or current starts from positive
charge and ends on negative charge.

12. BEHAVIOR OF MAGNETIC MATERIALS
 The materials which strongly attract a piece of iron are known as magnetic
materials or magnets
 The magnetic property of a material arises due to the magnetic moment or magnetic
dipole of materials
 Materials which are magnetized by the application of an external magnetic field are
known as magnetic materials
 Magnetic materials are categorized as magnetically hard, or magnetically soft
materials.
- Magnetically soft materials are easily magnetized but the induced magnetism is
usually temporary
- Like magnetically soft materials, magnetically hard materials can be
magnetized by a strong external magnetic field, such as those generated by an
electromagnet

13. Magnetic materials can be roughly classified into three main groups in
accordance with their relative permeability 𝜇r values
A material is said to be

14. CLASSIFICATION OF MAGNETIC MATERIALS

16. Anti ferro magnetic material
Atomic moments in antiparallel way
Net magnetic moment is zero
Affected only slightly by external field
Ex: Nickel oxide, manganese oxide
Ferri magnetic material
Antiparallel alignment of adjacent atomic moments but the moments are
not same
Large response for external field
Low conductivity than semiconductors
Ex:Nickel ferrite, Nickel zinc ferrite
Super Magnetic Material
Ferro and Non ferro magnetic materials

17. BOUNDARY CONDITIONS FOR MAGNETOSTATIC FIELDS
In homogeneous media, electromagnetic quantities vary smoothly and continuously. At a boundary between
dissimilar media, however, it is possible for electromagnetic quantities to be discontinuous.
Continuities and discontinuities in fields can be described mathematically by boundary conditions and used
to constrain solutions for fields away from these boundaries.
When magnetic field enter from one medium to another medium, there may be discontinuity in the
magnetic field, which can be explained by magnetic boundary condition
To study the conditions of H and B at the boundary, both the vectors are resolved into two components
(i) Tangential to the boundary (Parallel to boundary)
(ii) Normal to the boundary (Perpendicular to boundary)
These two components are resolved or derived using Ampere’s law and Gauss’s law
Consider two isotropic and homogeneous linear materials at the boundary with different permeabilities 𝜇1
and 𝜇2
Consider a rectangular path and gaussian surface to determine the boundary conditions

19. (i) Boundary conditions for Tangential Component
According to Ampere’s Circuital law 𝐻 ⋅ 𝑑𝐿 = 𝐼
𝑎
𝑎
𝐻. 𝑑𝐿 = 𝑎
𝑏
𝐻. 𝑑𝐿 + 𝑏
𝑐
𝐻. 𝑑𝐿 + 𝑐
𝑑
𝐻. 𝑑𝐿 + 𝑑
𝑎
𝐻. 𝑑𝐿 = I= Iencl ∆𝑤
= K ∆𝑤 ; K=Surface current normal to the Path (∆𝑠)
Here rectangular path height = ∆ℎ 𝑎𝑛𝑑 𝑊𝑖𝑑𝑡ℎ = ∆𝑤 ∆ℎ
K ∆𝑤 = Htan1(∆𝑤) + HN1(
∆ℎ
2
) + HN2(
∆ℎ
2
) – Htan2(∆𝑤) – HN2(
∆ℎ
2
) – HN2(
∆ℎ
2
)
At the boundary , ∆ℎ = 0 (∆ℎ/2- ∆ℎ/2)
K ∆𝑤 = Htan1(∆𝑤) – Htan2(∆𝑤)
In vector form,
𝐇 tan1 - 𝐇 tan2 = 𝐊 X 𝐚 N12
K = Htan1– Htan2

20. For 𝐵, the tangential component can be related with Permeabilities of two media
B = 𝜇 H, B tan1 = 𝜇1Htan1 & B tan2 = 𝜇2 H tan2
∴
B tan1
𝜇1
= Htan1 and
B tan2
𝜇2
= Htan2
B tan1
𝝁𝟏
-
B tan𝟐
𝝁2
= K
Special Case :
The boundary is of free of current then media are not conductor, so K = 0
Htan1 = Htan2
For tangential component of 𝐵 = 𝜇 𝐻 , 𝐻 = 𝐵/ 𝜇
B tan1
𝜇1
-
B tan2
𝜇2
= 0
B tan1
B tan2
=
𝜇1
𝜇2
=
𝜇𝑟1
𝜇𝑟2

21. (ii) Boundary Conditions for Normal Component
Closed Gaussian surface in the form of circular cylinder is consider to find the normal
component of 𝐵
According to Gauss’s law for magnetic field
𝑆
𝐵. 𝑑𝑠 = 0
The surface integral must be evaluated over 3 surfaces (Top, bottom and Lateral)
𝑆
𝐵. 𝑑𝑠 + 𝑆
𝐵. 𝑑𝑠 + 𝑆
𝐵. 𝑑𝑠 = 0
Top Bottom Lateral
At the boundary, ∆ℎ = 0,so only top and bottom surfaces contribute in the surface
integral
The magnitude of normal component of 𝐵 is BN1 and BN2
For top surface 𝑇𝑂𝑃
𝐵. 𝑑𝑠 = 𝑇𝑂𝑃
BN1 𝑑𝑠
= BN1 𝑇𝑂𝑃
𝑑𝑠
= BN1 ∆𝑠

22. For bottom surface 𝐵𝑜𝑡𝑡𝑜𝑚
𝐵. 𝑑𝑠 = 𝐵𝑜𝑡𝑡𝑜𝑚
BN2 𝑑𝑠
= BN2 𝐵𝑜𝑡𝑡𝑜𝑚
𝑑𝑠
= BN2 −∆𝑠 = − BN2 ∆𝑠
For Lateral surface 𝐿𝑎𝑡𝑒𝑟𝑎𝑙
𝐵. 𝑑𝑠 = 𝐵 𝐿𝑎𝑡𝑒𝑟𝑎𝑙
𝑑𝑠
= 𝐵 . ∆ℎ = 0
∴ BN1 ∆𝑠 − BN2 ∆𝑠 = 0
BN1 ∆𝑠 = BN2 ∆𝑠
BN1 = BN2
Thus the normal component of 𝐵 is continuous at the boundary
we know that 𝐵 = 𝜇 𝐻
For medium 1and 2
𝜇1 𝐻1N1 = 𝜇2 𝐻2N2

23. 𝐻1N1
𝐻2N2
=
𝜇2
𝜇1
=
𝜇r2
𝜇 r1
Hence the normal component of 𝐻 is not continuous at the boundary.
The field strength in two medias are inversely proportional to their relative
permeabilities

24. Thank you