Comprehensive Study Notes on Magnetic Materials

Fundamentals of Magnetic Materials

Magnets possess specific geometric and magnetic properties that define their interactions and field strengths. In the context of a permanent magnet, the physical dimensions are distinguished between the geometric and magnetic lengths.

• Geometric Length ($GL$): The actual physical length of the magnet from one physical end to the other.
• Magnetic Length ($ML$ or $2l$): The distance between the two poles of the magnet. Because the poles are located slightly inside the physical ends, the magnetic length is shorter than the geometric length.
• Relationship Formula:$ML = 0.84 \times GL$$ML \approx \frac{5}{6} GL$

Magnetic Dipole Moment and Pole Strength

A magnet acts as a magnetic dipole, consisting of two poles (North and South) of equal and opposite strength.

• Pole Strength ($m$): A measure of the strength of a magnetic pole. Its unit is Ampere-meter ($A \cdot m$).
• Magnetic Dipole Moment ($M$): A vector quantity representing the magnet's strength and orientation. It is defined as the product of the pole strength and the magnetic length.   $M = m(2l)$   The unit for Magnetic Dipole Moment is Ampere-square meter ($A \cdot m^2$).

Comparison with Electric Dipole

An electric dipole consists of two charges, $-q$ and $+q$, separated by a distance $d$.

• Electric Dipole Moment ($p$):$p = qd$   where $q$ is the charge and $d$ is the separation distance.

Resultant Magnetic Dipole Moment

When two magnets are combined or oriented at an angle, the resultant magnetic dipole moment ($M_R$) is calculated using vector addition.

• General Formula for Two Dipoles at Angle $\theta$:$M_R = \sqrt{M_1^2 + M_2^2 + 2M_1M_2 \cos(\theta)}$

• Specific Case ($\theta = 60^\circ$):   If $M_1 = M_2 = M$ and $\theta = 60^\circ$$M_R = \sqrt{M^2 + M^2 + 2M^2 \cos(60^\circ)}$

• Specific Case ($\theta = 120^\circ$):   If $M_1 = M_2 = M$ and $\theta = 120^\circ$$M_R = \sqrt{M^2 + M^2 + 2M^2 \cos(120^\circ)}$

Effects of Cutting a Magnet

Cutting a magnet changes its dipole moment depending on the orientation of the cut relative to the magnetic axis.

Vertical (Transverse) Cutting

If a magnet is cut vertically into $n$ equal parts:

• Pole Strength ($m$): Remains fixed/unchanged.
• New Length ($(2l)'$): The length of each piece becomes $(2l)' = \frac{2l}{n}$.
• New Dipole Moment ($M'$):$M' = m(2l)' = m\left(\frac{2l}{n}\right) = \frac{M}{n}$

Horizontal (Longitudinal) Cutting

If a magnet is cut horizontally into $n$ equal parts:

• New Pole Strength ($m'$): The pole strength is divided by the number of slices, $m' = \frac{m}{n}$.
• Length ($2l$): The magnetic length remains fixed/unchanged.
• New Dipole Moment ($M'$):$M' = m'(2l) = \left(\frac{m}{n}\right)(2l) = \frac{M}{n}$

Comparative Study of Dipoles

Calculations for magnetic fields ($B$) and electric fields ($E$) follow analogous mathematical structures. Let $r$ be the distance and $\theta$ be the angle from the dipole axis.

Constants

• Magnetic Constant ($C$): $C = \frac{\mu_0}{4\pi}$
• Electric Constant ($k$): $k = \frac{1}{4\pi\epsilon_0}$

Axial and Equatorial Fields

General Point $(r, \theta)$

For a point at distance $r$ and angle $\theta$ from the dipole:

• Net Magnetic Field ($B_{net}$):$B_{net} = \frac{CM}{r^3} \sqrt{1 + 3\cos^2(\theta)}$   The direction of the net field relative to the position vector is given by $\tan(\phi) = \frac{\tan(\theta)}{2}$.

• Net Electric Field ($E_{net}$):$E_{net} = \frac{kp}{r^3} \sqrt{1 + 3\cos^2(\theta)}$

Magnetic Dipoles in Uniform Fields

When a magnetic dipole (moment $M$) is placed in a uniform magnetic field ($B$), it experiences torque and possesses potential energy.

• Force ($F_{net}$): In a uniform field, the net force is zero ($F_{net} = 0$).
• Torque ($\tau$): The torque acts to align the dipole with the field.   $\tau = \mathbf{M} \times \mathbf{B}$$\tau = MB \sin(\theta)$
• Potential Energy ($U$):$U = -\mathbf{M} \cdot \mathbf{B} = -MB \cos(\theta)$
• Work Done ($W$): To rotate a dipole from angle $\theta_1$ to $\theta_2$:   $W_{\theta_1 \rightarrow \theta_2} = MB(\cos(\theta_1) - \cos(\theta_2))$

Equilibrium States

• Stable Equilibrium: Occurs when $\theta = 0^\circ$. Potential energy is at its minimum ($U_{min} = -MB$).
• Unstable Equilibrium: Occurs when $\theta = 180^\circ$. Potential energy is at its maximum ($U_{max} = MB$).

Time Period of Oscillation

If the dipole is slightly displaced from equilibrium, it undergoes simple harmonic motion with a time period ($T$): $T = 2\pi \sqrt{\frac{I}{MB}}$ Where $I$ is the Moment of Inertia of the magnet.

Properties of Magnetic Field Lines

1. Magnetic field lines always form closed loops.
   • Outside the magnet: Lines travel from the North pole to the South pole.
   • Inside the magnet: Lines travel from the South pole to the North pole.
2. A tangent drawn at any point on a field line provides the direction of the magnetic field ($B$) at that point.
3. Crowded regions of field lines indicate a strong field, while sparsely spaced lines indicate a weak field.

Atomic Basis of Magnetism

Magnetism in matter originates at the atomic level due to the motion of electrons.

1. Matter is made up of atoms.
2. When an electron revolves around the nucleus of an atom, it constitutes a current loop.
3. This current loop behaves as an atomic dipole with a magnetic moment $M = IA$.

Classification based on Atomic Spin

• Diamagnetic Materials: In certain materials, all electrons are paired. The magnetic moments of paired electrons cancel each other out, resulting in a net magnetic moment of zero ($M_{net} = 0$).
• Paramagnetic and Ferromagnetic Materials: In some materials, electrons are unpaired, leading to individual atoms having a net magnetic moment ($M_{atomic} \neq 0$).

Magnetization and Magnetic Intensity

In the absence of an external magnetic field ($H$), the atomic dipoles in paramagnetic or ferromagnetic materials are randomly oriented, making the macroscopic net magnetic moment zero.

When an external magnetic field is applied, a torque acts on the atomic dipoles, trying to align them in the direction of the field, resulting in $M_{net} \neq 0$.

Magnetization ($I$)

Magnetization is defined as the net magnetic moment per unit volume. $I = \frac{M}{V}$ The unit is Ampere per meter ($A \cdot m^{-1}$).

Magnetic Intensity ($H$)

Consider a solenoid carrying a current $i$ with a soft iron core placed inside. The magnetic field inside the solenoid depends on two factors:

1. External Factor: The current ($i$) creating the magnetizing field ($H$).
2. Internal Factor: The alignment of atomic dipoles ($I$).

Net Magnetic Field ($B_{net}$) inside the core: $B \propto (H + I)$$B = \mu_0(H + I)$ Units: The unit of $H$ is the same as the unit of $I$, which is Ampere per meter ($A \cdot m^{-1}$).

Magnetic Susceptibility and Permeability

• Magnetic Susceptibility ($\chi$): This represents how easily a material can be magnetized. It is the ratio of magnetization to magnetic intensity.   $I \propto H \implies I = \chi H$$\chi$ is a dimensionless quantity.

• Total Magnetic Field ($B_m$) in a Medium:$B_m = \mu_0(H + I)$   Substituting $I = \chi H$:   $B_m = \mu_0(H + \chi H) = \mu_0(1 + \chi)H$

• Permeability Relations:

  • Absolute Permeability ($\mu_m$): $B_m = \mu_m H \implies \mu_m = \mu_0(1 + \chi)$
  • Relative Permeability ($\mu_r$): The ratio of the permeability of the medium ($\mu_m$) to the permeability of vacuum ($\mu_0$).     $\mu_r = \frac{\mu_m}{\mu_0} = \frac{B_m}{B_0} = 1 + \chi$