CHAPTER 4
Magnetic Properties of Matter
4.1. The Magnetic Dipole
As we discussed in the previous chapter, we
can approximate a magnetic dipole as a small
current-carrying loop located far from our point
of interest. For most circumstances, we can ima-
gine a magnetic dipole as a tiny bar magnet
with a North Pole and a South Pole. The di-
pole can be taken to be the fundamental source
of a magnetic field; there will be no other sources
that can be broken down further. This is unlike
the electric field, where the fundamental (ideal)
source is a point charge. Such point "magnetic
charges" do not exist, making the dipole a smal-
lest unit. This is reflected in the Maxwell equa-
tion: ∇ • B⃗ =0
Let us imagine some circular current-carrying F IGURE 4.1.1: Current carrying loop as a magnetic di-
loop with an area A, radius r, and a current I, pole
as seen in Figure1 4.1.1. The magnetic potential
due to the loop at some point P (r, θ, ϕ) located at a distance r from the loop is given by
ˆ
⃗ µ0 I ⃗
A= dl
4π L r
For the case r ≫ a where the point P is far from the loop, the integral resolves to
2
⃗ = µ 0 Iπa
A sin θ ϕ̂
4π r2
Recognizing that the magnetic dipole moment can be written as m = Iπa2 and that the sin θ term
indicates the presence of a cross product, we can rewrite the above as
⃗ × r̂
⃗ = µ0 m
A (4.1.1)
4π r2
1
Image taken from Sadiku, M.N.O. (2018). Elements of electromagnetics (7th ed.). Oxford University Press. Page 363.
94
, CHAPTER 4. MAGNETIC PROPERTIES OF MATTER 95
From here we can get the magnetic field intensity (in spherical coordinates) as
µ m
⃗ =∇×A
⃗ = 0
B 3
2 cos θ r̂ + sin θ θ̂ (4.1.2)
4π r
As we can see, the magnetic field due to the dipole has the same 1/r3 dependence as with the electric
field due to an electric dipole.
4.2. Magnetization in Materials
The discussion of magnetism in materials is very similar to that of electric fields in dielectrics. Thus
we will be covering similar concepts and ideas.
Fundamentally, all materials can be broken down to a collection of atoms which has one or more
negatively charged electrons rotating about a positively charged nucleus. Since a charge is moving,
we can think of this as a tiny electric current moving in a circular loop, which would produce a small
magnetic field. There is also an additional magnetic field produced by the spin of an electron. The
concept of spin will be covered in more detail in the future Quantum Mechanics course. For now,
all we need to keep in mind is that this property also involves the motion of the electron charge and
thus creates another tiny magnetic field. The total tiny magnetic dipole moment can be represented
by m = Ib A, where Ib is the current representing the charged motion of the electron and A is the
area of its orbit.
F IGURE 4.2.1: Orientation of magnetic dipoles (a) normally, and (b) in the
⃗
presence of an external magnetic field B
In general, the dipoles inside a material are oriented in random directions, as seen in Figure2 4.2.1
(a). Thus the overall magnetic moment of the material (sum of all magnetic moments) is effectively
zero. However, when an external magnetic field is applied to the material (seen in part (b) of the
figure), then the dipole moments all align (or try to align) in the same direction. This will now impart
a non-zero overall magnetic moment to the material.
⃗ as the total magnetic dipole moment per unit volume
We now define the magnetism of a material (M)
inside a material. Its SI unit is A/m.
Pn
⃗j
j=1 m
⃗ = Lt
M (4.2.1)
△τ→0 △τ
⃗ j , within an infinitesimally small volume
where we assume there are n atoms, each with a moment m
△τ.
2
Ibid. Page 369.
Magnetic Properties of Matter
4.1. The Magnetic Dipole
As we discussed in the previous chapter, we
can approximate a magnetic dipole as a small
current-carrying loop located far from our point
of interest. For most circumstances, we can ima-
gine a magnetic dipole as a tiny bar magnet
with a North Pole and a South Pole. The di-
pole can be taken to be the fundamental source
of a magnetic field; there will be no other sources
that can be broken down further. This is unlike
the electric field, where the fundamental (ideal)
source is a point charge. Such point "magnetic
charges" do not exist, making the dipole a smal-
lest unit. This is reflected in the Maxwell equa-
tion: ∇ • B⃗ =0
Let us imagine some circular current-carrying F IGURE 4.1.1: Current carrying loop as a magnetic di-
loop with an area A, radius r, and a current I, pole
as seen in Figure1 4.1.1. The magnetic potential
due to the loop at some point P (r, θ, ϕ) located at a distance r from the loop is given by
ˆ
⃗ µ0 I ⃗
A= dl
4π L r
For the case r ≫ a where the point P is far from the loop, the integral resolves to
2
⃗ = µ 0 Iπa
A sin θ ϕ̂
4π r2
Recognizing that the magnetic dipole moment can be written as m = Iπa2 and that the sin θ term
indicates the presence of a cross product, we can rewrite the above as
⃗ × r̂
⃗ = µ0 m
A (4.1.1)
4π r2
1
Image taken from Sadiku, M.N.O. (2018). Elements of electromagnetics (7th ed.). Oxford University Press. Page 363.
94
, CHAPTER 4. MAGNETIC PROPERTIES OF MATTER 95
From here we can get the magnetic field intensity (in spherical coordinates) as
µ m
⃗ =∇×A
⃗ = 0
B 3
2 cos θ r̂ + sin θ θ̂ (4.1.2)
4π r
As we can see, the magnetic field due to the dipole has the same 1/r3 dependence as with the electric
field due to an electric dipole.
4.2. Magnetization in Materials
The discussion of magnetism in materials is very similar to that of electric fields in dielectrics. Thus
we will be covering similar concepts and ideas.
Fundamentally, all materials can be broken down to a collection of atoms which has one or more
negatively charged electrons rotating about a positively charged nucleus. Since a charge is moving,
we can think of this as a tiny electric current moving in a circular loop, which would produce a small
magnetic field. There is also an additional magnetic field produced by the spin of an electron. The
concept of spin will be covered in more detail in the future Quantum Mechanics course. For now,
all we need to keep in mind is that this property also involves the motion of the electron charge and
thus creates another tiny magnetic field. The total tiny magnetic dipole moment can be represented
by m = Ib A, where Ib is the current representing the charged motion of the electron and A is the
area of its orbit.
F IGURE 4.2.1: Orientation of magnetic dipoles (a) normally, and (b) in the
⃗
presence of an external magnetic field B
In general, the dipoles inside a material are oriented in random directions, as seen in Figure2 4.2.1
(a). Thus the overall magnetic moment of the material (sum of all magnetic moments) is effectively
zero. However, when an external magnetic field is applied to the material (seen in part (b) of the
figure), then the dipole moments all align (or try to align) in the same direction. This will now impart
a non-zero overall magnetic moment to the material.
⃗ as the total magnetic dipole moment per unit volume
We now define the magnetism of a material (M)
inside a material. Its SI unit is A/m.
Pn
⃗j
j=1 m
⃗ = Lt
M (4.2.1)
△τ→0 △τ
⃗ j , within an infinitesimally small volume
where we assume there are n atoms, each with a moment m
△τ.
2
Ibid. Page 369.
