Magnetism and Magnetic Materials

Properties of Permanent Magnets and Magnetic Materials

• Magnetic Pole Strength ($m$):

  • Measured in Ampere-meters ($A\,m$).

  • Represents the strength of a magnetic pole.

• Dimensions of a Magnet:

  • Geometric Length ($GL$): The actual physical length of the magnet from end to end.

  • Magnetic Length ($ML$): The distance between the two poles (North and South) of the magnet.

  • Relationship: The magnetic length is shorter than the geometric length due to the poles being located slightly inside the material.

    • $ML = 0.84 \times GL$

    • $ML = \frac{5}{6} GL$

• Magnetic Dipole Moment ($M$):

  • It is a vector quantity defined as the product of the pole strength ($m$) and the magnetic length ($2l$).

  • Formula: $M = m \times (2l)$

  • Unit: Ampere-square meters ($A\,m^2$).

  • Direction: Always directed from the South Pole ($S$) to the North Pole ($N$).

Comparison of Electric and Magnetic Dipoles

• Electric Dipole:

  • Consists of two charges, $-q$ and $+q$, separated by a distance $d$.

  • Electric Dipole Moment ($p$): $p = q \times d$.

  • Unit: Coulomb-meters ($C\,m$).

• Resultant Magnetic Dipole Moment ($M_R$):

  • When two magnetic moments $M_1$ and $M_2$ are placed at an angle $\theta$ to each other, the resultant moment is found via vector addition.

  • General Formula: $M_R = \sqrt{M_1^2 + M_2^2 + 2M_1 M_2 \cos(\theta)}$

  • Special Case: If two identical magnets ($M_1 = M_2 = M$) are placed perpendicular to each other ($\theta = 90^\circ$):

    • $M_R = \sqrt{M^2 + M^2} = \sqrt{2}M$

Effects of Cutting a Magnet

• Case 1: Cutting Transverse to the Length:

  • The magnet is cut into $n$ equal pieces perpendicular to its axis.

  • Pole Strength ($m$): Remains fixed/unchanged.

  • Length ($l'$): Becomes $\frac{2l}{n}$.

  • New Magnetic Moment ($M'$): $M' = m \left(\frac{2l}{n}\right) = \frac{M}{n}$.

• Case 2: Cutting Parallel to the Length (Longitudinal):

  • The magnet is cut into $n$ equal pieces along its axis.

  • Length ($2l$): Remains fixed/unchanged.

  • Pole Strength ($m'$): Becomes $\frac{m}{n}$.

  • New Magnetic Moment ($M'$): $M' = \left(\frac{m}{n}\right) \times 2l = \frac{M}{n}$.

Magnetic Dipole Field Calculations

• Magnetic Field Intensity ($B$):

  • Constant $C$ is often used in the transcript as a proxy for $\frac{\mu_0}{4\pi}$.

• Axial Point:

  • The field at a point on the axis of the dipole.

  • Formula: $B_{axial} = \frac{2CM}{r^3}$

  • Electric Analogue: $E_{axial} = \frac{2kP}{r^3}$

• Equatorial/General Point:

  • The field at a point at distance $r$ and angle $\theta$ from the dipole center.

  • Formula: $B_{net} = \frac{CM}{r^3} \sqrt{1 + 3 \cos^2(\theta)}$

  • Electric Analogue: $E = \frac{kP}{r^3} \sqrt{3 \cos^2(\theta) + 1}$

• Direction of the Field (Angle $\alpha$):

  • The angle between the resultant magnetic field and the position vector $r$.

  • Formula: $\tan(\alpha) = \frac{\tan(\theta)}{2}$

Behavior in Uniform Fields

• In a Uniform Magnetic Field ($B$):

  • Net Force ($F_{net}$): $0$ (The forces on both poles cancel out).

  • Torque ($\tau$): $\tau = M \times B = MB \sin(\theta)$.

  • Potential Energy ($U$): $U = -M \cdot B = -MB \cos(\theta)$.

• Equilibrium States:

  • Stable Equilibrium: Occurs when $\theta = 0^\circ$. Potential energy is at its minimum ($U = -MB$). The dipole is parallel to the field.

  • Unstable Equilibrium: Occurs when $\theta = 180^\circ$. Potential energy is at its maximum ($U = +MB$). The dipole is anti-parallel to the field.

• Work Done ($W$):

  • Work required to rotate a magnet from angle $\theta_1$ to $\theta_2$.

  • Formula: $W = MB (\cos(\theta_1) - \cos(\theta_2))$

• Time Period of Oscillation ($T$):

  • For small oscillations in a uniform field:

  • Formula: $T = 2\pi \sqrt{\frac{I}{MB}}$

  • Where $I$ is the Moment of Inertia of the magnet.

Magnetic Susceptibility and Permeability

• Magnetization ($I$):

  • The degree to which a material becomes magnetized when placed in an external field.

  • It is influenced by external factors and dimensions.

  • Relation: $I = \chi H$

  • Stronger external magnetic fields ($H$) lead to higher magnetization ($I$).

• Magnetic Susceptibility ($\chi$):

  • Represents how easily a substance can be magnetized.

  • Dimensionless quantity related to the material's response to an external field ($H$).

• Permeability Relations:

  • Absolute Permeability ($\mu$): $\mu = \mu_0 \mu_r$.

  • Vacuum Permeability ($\mu_0$): In vacuum, $I = 0$, so $B = \mu_0 H$.

  • Relative Permeability ($\mu_r$): $\mu_r = \frac{\mu}{\mu_0} = 1 + \chi$.

  • Net Magnetic Field ($B_{net}$): $B_{net} = \mu_0 (H + I)$.

Properties of Magnetic Field Lines

1. Closed Loops: Field lines always form continuous closed loops. Inside the magnet, they travel from the South Pole to the North Pole. Outside the magnet, they travel from the North Pole to the South Pole.

2. Direction: A tangent drawn at any point on a field line provides the direction of the magnetic field at that point.

3. Field Strength: Crowded (dense) field lines indicate a strong magnetic field region, while spread-out lines indicate a weak field region.

Microscopic Origins of Magnetism

• Atomic Level Activity:

  • Matter is made of atoms.

  • Electrons revolve around the nucleus, constituting a tiny current loop.

  • This current loop behaves as a magnetic dipole (Atomic Dipole).

  • Atomic Magnetic Moment: Represented as $M_{atom}$.

• Effect of External Magnetic Field ($H$):

  • When a sample is placed in an external field, torque acts on these atomic dipoles, attempting to align them in the direction of the field ($\tau = M_{net} \times B$).

• Definition of Magnetization ($I$):

  • Defined as the Net Magnetic Moment per unit Volume.

  • Formula: $I = \frac{M}{V}$

  • Unit: Ampere per meter ($A\,m^{-1}$).

Types of Magnetic Materials

• Diamagnetism:

  • Occurs in materials where all electrons are paired.

  • The magnetic moments of paired electrons cancel each other out.

  • Net Atomic Moment: $M_{atom} = 0$.

• Paramagnetism and Ferromagnetism:

  • Occurs in materials where electrons are unpaired.

  • Each atom has a non-zero net magnetic moment ($M_{atom} \neq 0$).

  • In the Absence of External Field: These moments are randomly oriented due to thermal agitation, so the net magnetic moment of the bulk material is still zero ($M_{net} = 0$).

The Magnetizing Field (H)

• Solenoid Example:

  • Consider a solenoid carrying current with a soft iron core (ferromagnetic) inside.

  • The magnetic field inside is due to two factors:

    1. External Factor ($H$): The magnetizing field produced by the current in the solenoid ($H$).

    2. Internal Factor ($I$): The alignment of atomic dipoles within the iron core (Magnetization).

  • Net Magnetic Field ($B$): $B = \mu_0 (H + I)$.

  • Units: Both $H$ (Magnetic Intensity) and $I$ (Magnetization) have the unit Ampere/meter ($A\,m^{-1}$).