- describes how different magnetic materials behave in the presence of external magnetic field - presents the difference between electric circuit analysis and magnetic circuit analysis.

1. Kailash Karki, nec Electromagnetic Fields & Waves
10. Magnetic Force and Material Media

2. Kailash Karki, nec Electromagnetic Fields & Waves
10.1:Magnetic Force
• The electric field causes a force to be exerted on a charge which may be either stationary or in motion
• Steady magnetic field is capable of exerting a force only on moving charges
10.1.1:Force on a Moving Charge
• In an electric field, the force on a charged particle is given by
𝑭 = 𝑄𝑬
• The force is in the same direction as the electric field intensity(for a positive charge) and is directly
proportional to both 𝑄 𝑎𝑛𝑑 𝑬.
• A charged particle in motion in a magnetic of flux density B is found experimentally to experience a force
𝑭 = 𝑄𝒗 ∗ 𝑩 [𝑣 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦]
• Force due to magnetic field can never change the magnitude of the particle velocity but can change the
direction.
• The force on a moving particle arising from combined effect of electric and magnetic fields is obtained by
superposition:
𝑭 = 𝑄(𝑬 + 𝒗 ∗ 𝑩)
• This equation is known as the Lorentz force equation.

3. Kailash Karki, nec Electromagnetic Fields & Waves
10.1.2: Force on a Differential Current Element
• Differential element of charge consists of a large number of very small, discrete charges occupying a volume
• The force on a charged particle moving through a steady magnetic field may be written as the differential
force exerted on a differential element of charge,
𝑑𝑭 = 𝑑𝑄 𝒗 ∗ 𝑩 = 𝜌 𝑣 𝑑𝑣 𝒗 ∗ 𝑩 [since, dQ = 𝜌 𝑣 𝑑𝑣 ]
Or, 𝑑𝑭 = 𝑱 ∗ 𝑩 𝑑𝑣[sinc𝒆, 𝑱 = 𝜌 𝑣 𝒗]
Or, 𝑑𝑭 = 𝐼 𝒅𝑳 ∗ 𝑩 [𝑱 𝑑𝑣 = 𝐼 𝒅𝑳 ]
• For a closed circuit carrying a current 𝐼, the total magnetic force is
𝑭 = ර 𝐼 𝒅𝑳 ∗ 𝑩 = 𝐼 𝑳 ∗ 𝑩
• The magnitude of the force is given by the equation, 𝐹 = 𝐵𝐼𝐿𝑠𝑖𝑛𝜃

4. Kailash Karki, nec Electromagnetic Fields & Waves
10.1.3: Force Between Differential Current Elements
• The magnetic field at point 2 due to a current element at point 1 is found to be
𝑑𝑯 𝟐 =
𝐼1 𝑑𝒍 𝟏∗𝑎 𝑅12
(4𝜋)𝑅12
2 , 𝑑𝑩 𝟐 =
μ0 𝐼1 𝑑𝒍 𝟏∗𝑎 𝑅12
(4𝜋)𝑅12
2 𝑠𝑖𝑛𝑐𝑒 𝐵 = μ0 𝐻
• Now, the differential force on a differential current element is
𝑑𝑭 = 𝐼 𝑑𝒍 ∗ 𝑩, 𝑑𝑭2 = 𝐼2 𝑑𝒍2 ∗ 𝑩2
• Differential amount of differential force at current element 2 is,
𝑑(𝑑𝑭2) = 𝐼2 𝑑𝒍2 ∗ 𝑑𝑩2
𝑑(𝑑𝑭2) = 𝐼2 𝑑𝒍2 ∗
μ0 𝐼1 𝑑𝒍 𝟏 ∗ 𝑎 𝑅12
(4𝜋)𝑅12
2 =
𝜇0 𝐼1 𝐼2
4𝜋
𝑑𝑙2 ∗ (𝑑𝑙1 ∗ 𝑎 𝑅12
)
𝑅12
2
• To calculate the total force between two line current, we must integrate over the length of each wire.
𝐹2 = ‫׭‬ 𝑑 𝑑𝐹2 =
𝜇0 𝐼1 𝐼2
4𝜋
‫׭‬
𝑑𝑙2∗(𝑑𝑙1∗𝑎 𝑅12)
𝑅12
2
𝐼1 𝑑𝑙1 𝐼2 𝑑𝑙2𝑅12
21

5. Kailash Karki, nec Electromagnetic Fields & Waves
10.2: Nature of Magnetic Materials
10.2.1: Magnetic Dipole Moment
• The magnetic moment is a vector quantity used to measure the tendency of an object to interact with an
external magnetic field
• It determines the torque object will experience in an external magnetic field.
• Magnetic dipole moments have dimensions of current times area.
• Sources of magnetic dipole moment:
(a) Orbiting Electrons (𝑚 𝑜𝑟𝑏)
(b) Electron Spin(𝑚 𝑠𝑝𝑖𝑛)
• Characteristics of magnetic materials:
Diamagnetic Paramagnetic Ferromagnetic
𝑚 𝑜𝑟𝑏 + 𝑚 𝑠𝑝𝑖𝑛 = 0 𝑚 𝑜𝑟𝑏 + 𝑚 𝑠𝑝𝑖𝑛 = 𝑠𝑚𝑎𝑙𝑙 𝑚 𝑜𝑟𝑏 ≫ 𝑚 𝑠𝑝𝑖𝑛
No magnetic dipole
moment in the absence of
externa field.
Small magnetic dipole
moment in the absence of
external field.
Relatively large dipole
moment in the absence of
external field.

6. Kailash Karki, nec Electromagnetic Fields & Waves
10.2.2:Magnetization
• The process of making a substance temporarily or permanently magnetic as by insertion in a magnetic field
• Magnetization or magnetic polarization is a vector field that expresses the density of permanent or induced
magnetic dipole moments in a magnetic material.
• The current produced by the orbital electrons and electron spin is due to the
movement of bound charges
• Magnetic dipole moment(m) is defined as the product of the area of small plane loop and
magnitude of circulating current and direction is perpendicular to the plane of current loop.
𝐹𝑜𝑟 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑖𝑎𝑙 𝑠𝑢𝑟𝑓𝑎𝑐𝑒, 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑚𝑜𝑚𝑒𝑛𝑡(𝑚) = 𝐼 𝑏 𝑑𝑆
• If there are n magnetic dipoles per unit volume , then the total magnetic dipole moment is given
by, 𝑚 𝑡𝑜𝑡𝑎𝑙 = σ𝑖=1
𝑛
𝑚𝑖
• Magnetization(M) is defined as the total magnetic dipole moment 𝑚 𝑡𝑜𝑡𝑎𝑙 per unit volume ,
𝑀𝑎𝑔𝑛𝑒𝑡𝑖𝑧𝑎𝑡𝑖𝑜𝑛(𝑀) = lim
∆𝑣→0
1
Δ𝑣
෍
𝑖=1
𝑛
𝑚𝑖

7. Kailash Karki, nec Electromagnetic Fields & Waves
Derivation [𝑩 = 𝜇0 𝑯 + 𝑴]
• Current due to movement of bound charges (𝐼 𝑏) is responsible for Magnetization, so ampere’s law can be
written as , 𝐼 𝑏 = ‫ׯ‬ 𝑀. 𝑑𝑙
• Writing Ampere’s law in terms of total current, ‫ׯ‬ 𝐻. 𝑑𝑙 = 𝐼 𝑇
‫ׯ‬ 𝐻. 𝑑𝑙 = 𝐼 + 𝐼 𝑏
[𝐼 𝑇 = 𝐼 + 𝐼 𝑏, 𝐼 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑑𝑢𝑒 𝑡𝑜 𝑓𝑟𝑒𝑒 𝑐ℎ𝑎𝑟𝑔𝑒𝑠, 𝐼 𝑏 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑑𝑢𝑒 𝑡𝑜 𝑏𝑜𝑢𝑛𝑑 𝑐ℎ𝑎𝑟𝑔𝑒𝑠]
Or, ‫ׯ‬ 𝐻. 𝑑𝑙 − 𝐼 𝑏 = 𝐼
Or, ‫ׯ‬
𝐵
μ0
. 𝑑𝑙 − ‫ׯ‬ 𝑀. 𝑑𝑙 = 𝐼
Or, ‫ׯ‬(
𝐵
μ0
− 𝑀). 𝑑𝑙 = 𝐼
Comparing with ‫ׯ‬ 𝐻. 𝑑𝑙 = 𝐼 , 𝐻 =
𝐵
𝜇0
− 𝑀
𝐵 = 𝜇0 𝐻 + 𝑀 𝑤𝑏/𝑚2

8. Kailash Karki, nec Electromagnetic Fields & Waves
10.2.3: Magnetic Susceptibility
• Magnetic susceptibility is the ratio of magnetization M (magnetic moment per unit volume) to the applied
magnetizing field intensity H. χ 𝑚 =
𝑀
𝐻
𝑜𝑟 𝑀 = χ 𝑚 𝐻
• We have, 𝐵 = 𝜇0 𝐻 + 𝑀
or, 𝐵 = 𝜇0 𝐻 + χ 𝑚 𝐻 = 𝜇0 𝐻 1 + χ 𝑚
𝑜𝑟, 𝐵 = 𝜇0 𝐻 𝜇 𝑟 where 𝜇 𝑟 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦
𝜇 𝑟 = 1 + χ 𝑚
Or, 𝐵 = 𝜇0μ 𝑟 𝐻
𝐵 = μ𝐻 [μ = 𝜇0μ 𝑟]

9. Kailash Karki, nec Electromagnetic Fields & Waves
10.3: Magnetic Boundary Condition
• Figure shows a boundary between two materials
with permeabilities 𝜇1 𝑎𝑛𝑑 𝜇2.
• We will use following laws to derive boundary
condition:
For Normal Component:
‫ׯ‬ 𝐵. 𝑑𝑆 = 0(𝐺𝑎𝑢𝑠𝑠 𝑙𝑎𝑤 𝑓𝑜𝑟 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑠𝑚)
For Tangential Component:
‫ׯ‬ 𝐻. 𝑑𝑙 = 𝐼 (𝐴𝑚𝑝𝑒𝑟𝑒′
𝑠 𝐶𝑖𝑟𝑐𝑢𝑖𝑡𝑎𝑙 𝑙𝑎𝑤)
• For Normal Component, For Tangential Component:

10. Kailash Karki, nec Electromagnetic Fields & Waves
10.4: Magnetic Circuits
• In magnetic circuits, we determine the magnetic fluxes and magnetic field intensities in various part of the
circuits. It is an important component in the deign of electrical machines.
• The common examples of magnetic circuits are transformers, generators, relays etc.
• The analogy between electric and magnetic circuit helps to ease the analysis .
Electric Circuit Magnetic Circuit
Electric Circuit Magnetic Circuit
1. Conductivity (σ) 1. Permeability(μ)
2.Field Intensity, E 2. Field Intensity, H
3. σ 𝐼 = 0 3. σ ψ = 0
4. 𝐼 = ‫׬‬ 𝐽. 𝑑𝑆 4. ψ = ‫׬‬ 𝐵. 𝑑𝑆
Reluctance: property of a material which opposes the creation of
magnetic flux in it.
Magneto-Motive Force(Mmf) : the external force required to set up
the magnetic flux lines within the magnetic material.
𝑀𝑚𝑓 𝐹 = 𝑁𝐼
N-Number of turns, I - Current

11. Kailash Karki, nec Electromagnetic Fields & Waves
References:
1. W H Hayt, J. A. (2014). Engineering Electromagnetics. New Delhi: McGraw Hill Education(India) Pvt. Ltd.