Magnetism and Magnetic Properties of Matter Comprehensive Study Guide

Hysteresis and Magnetic Properties

• Hysteresis Loop Progression:

  • Point O: The material starts in an unmagnetized state ($M = 0, H = 0, B = 0$).

  • Point A (Saturation Point): As the external magnetizing field ($H$) increases, the magnetic induction ($B$) increases and reaches a maximum value called the saturation point. This state is reached when all magnetic dipoles are aligned with the field.

  • Return Path (A to B): This process is not reversible. When the external field intensity ($H$) is reduced to zero, the material does not return to point O. Instead, it follows a path to point B.

  • Retentivity (Point B): The value of magnetic induction ($B$) remaining in the material when the magnetizing field ($H$) is reduced to zero. It represents the residual magnetism of the material ($B \neq 0, H = 0$).

  • Coercivity (Point C): To reduce the magnetic induction ($B$) to zero, a reverse magnetizing field must be applied. The magnitude of the reverse field ($H$) required to make $B = 0$ is called the coercivity ($B = 0, H \neq 0$).

  • Reverse Saturation (Point D): Further increase of the field in the reverse direction leads to saturation in that direction.

  • Closing the Loop (D to A through F): Reducing and then reversing the field again completes the hystereis loop through point F (where $B = 0$ in the opposite cycle).

Magnetic Formulae Sheet

• Magnetic Dipole Moment ($M$):

  • For a bar magnet: $M = m \times 2L$ (where $m$ is pole strength and $2L$ is magnetic length).

  • For a current-carrying loop: $M = I \times A$ (where $I$ is current and $A$ is area). Unit: $Am^2$.

• Magnetic Induction ($B$):

  • At any point: $B = \frac{\mu_0}{4\pi} \frac{M}{r^3} \sqrt{1 + 3\cos^2(\theta)}$.

  • At Axial Point: $B_{axis} = \frac{\mu_0}{4\pi} \frac{2M}{r^3}$.

  • At Equatorial Point: $B_{eq} = \frac{\mu_0}{4\pi} \frac{M}{r^3}$.

• Torque ($\tau$):

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

• Work and Potential Energy:

  • Work done in rotating a dipole: $W = U = MB [\cos(\theta_1) - \cos(\theta_2)]$.

  • If starting from equilibrium ($\theta_1 = 0^{\circ}$) to an angle ($\theta_2 = \theta$): $W = MB [1 - \cos(\theta)]$.

• Oscillation and Time Period:

  • Angular acceleration: $\alpha = \frac{\tau}{I} = \frac{MB \sin(\theta)}{I}$.

  • Time Period: $T = 2\pi \sqrt{\frac{I}{MB}}$.

Magnetic Materials and Definitions

• Pole Strength ($m$): Measured in Ampere-meter ($Am$). Dimension: $[AL]$.

• Magnetic Length ($2L$): The distance between the two poles of a magnet. Relationship with geometric length: $\text{Magnetic Length} = \frac{5}{6} \times \text{Geometric Length}$.

• Magnetization ($M_z$ or $I$): Defined as the net magnetic dipole moment per unit volume. $M_z = \frac{M_{net}}{\text{Volume}}$. Unit: $A/m$. Dimension: $[AL^{-1}]$.

• Magnetic Intensity ($H$): The magnetizing field, often related to the number of turns and current in a solenoid. $H = nI$. Unit: $A/m$.

• Magnetic Susceptibility ($\chi$): Relationship between magnetization and magnetic intensity: $M_z = \chi H$. High sensitivity means the object gets magnetized easily.

• Permeability ($\mu$):

  • $\mu = \mu_0 \mu_r$

  • $\mu_0$ = Permeability of free space.

  • $\mu_r$ = Relative permeability.

  • Relation: $\mu_r = 1 + \chi$.

• Total Magnetic Induction ($B$):

  • $B = \mu_0 (H + M_z) = \mu_0 (H + \chi H) = \mu_0 H (1 + \chi) = \mu_0 \mu_r H = \mu H$.

Classification of Magnetic Materials

• Diamagnetic Substances:

  • Weakly repelled in a magnetizing field.

  • Move from stronger to weaker parts of a non-uniform magnetic field.

  • Field lines are less dense inside the material compared to outside.

  • Properties are independent of temperature.

  • Examples: Bismuth, Copper, Gold, Mercury, Quartz, Alcohol, Hydrogen ($H_2$), Water.

  • Relative permeability \mu_r < 1, and susceptibility $\chi$ is negative.

• Paramagnetic Substances:

  • Weakly attracted towards a magnetic field.

  • Move from weaker to stronger parts of a non-uniform field.

  • Field lines are slightly more dense inside the material.

  • Magnetization increases as the material is cooled.

  • Examples: Aluminum, Platinum, Manganese, Oxygen, Copper Sulphate ($CuSO_4$).

  • Relative permeability \mu_r > 1, and susceptibility $\chi$ is positive and small.

• Ferromagnetic Substances:

  • Strongly attracted towards a magnetic field.

  • Move quickly from weaker to stronger parts of a non-uniform field.

  • Possess very large resultant magnetic moments due to "domain theory" (regions where spins are aligned parallelly).

  • Dependent on temperature; converts to paramagnetic at the Curie Temperature ($T_c$).

  • Examples: Iron ($Fe$), Cobalt ($Co$), Nickel ($Ni$), Gadolinium ($Gd$), and alloys like Alnico and Nipermag.

  • Relative permeability $\mu_r \gg 1$, and susceptibility $\chi$ is a vary large positive value.

Curie's Law

• Magnetization ($M_z$) is directly proportional to the external magnetic field ($B$) and inversely proportional to the absolute temperature ($T$).

• $M_z \propto \frac{B}{T} \rightarrow M_z = C \frac{B}{T}$, where $C$ is the Curie constant.

• Since $B \approx \mu_0 H$, we can state $M_z = \frac{C \mu_0 H}{T}$, leading to $\chi = \frac{C \mu_0}{T}$.

Atomic Magnetism and Revolving Electrons

• An electron revolving in a circular orbit of radius ($r$) with speed ($v$) constitutes a current ($I$).

• $I = \frac{e}{T} = \frac{e}{2\pi r / v} = \frac{ev}{2\pi r}$.

• Magnetic Dipole Moment ($M$): $M = I \times A = \frac{ev}{2\pi r} \times \pi r^2 = \frac{evr}{2}$.

• Angular Momentum ($L$): $L = m_e vr$.

• Relationship between $M$ and $L$: $\frac{M}{L} = \frac{e}{2m_e}$.

• Gyromagnetic Ratio: The ratio $\frac{M}{L} = \frac{e}{2m_e} = 8.8 \times 10^{10}\,\text{C/kg}$.

• Bohr Magneton ($\mu_B$): The smallest unit of magnetic moment.

  • $\mu_B = \frac{eh}{4\pi m_e} = 9.27 \times 10^{-24}\,Am^2$.

  • $h = \text{Planck's constant} = 6.63 \times 10^{-34}\,J\,s$.

Vector Addition of Dipole Moments

When two bar magnets with moments $M_1$ and $M_2$ are placed at an angle ($\theta$):

• $M_{net} = \sqrt{M_1^2 + M_2^2 + 2M_1 M_2 \cos(\theta)}$.

• If $M_1 = M_2 = M$:

  • For $\theta = 60^{\circ}$: $M_{net} = \sqrt{3}M$.

  • For $\theta = 90^{\circ}$: $M_{net} = \sqrt{2}M$.

  • For $\theta = 120^{\circ}$: $M_{net} = M$.

  • For Parallel ($\theta = 0^{\circ}$): $M_{net} = M_1 + M_2$.

  • For Antiparallel ($\theta = 180^{\circ}$): $M_{net} = M_2 - M_1$.

Example Problems: Bending Wires and Magnets

• Bending a wire of length $L$ into shapes:

  • L-shape (bent at center): If a magnet of moment $M$ is bent at its midpoint, the effective displacement between poles becomes $\sqrt{(\frac{L}{2})^2 + (\frac{L}{2})^2} = \frac{L}{\sqrt{2}}$. The new moment $M' = \frac{M}{\sqrt{2}}$.

  • Semi-circle: If a magnet of length $L$ is bent into a semi-circle of radius $R$, then $L = \pi R$, so $R = \frac{L}{\pi}$. The effective distance between poles is $2R = \frac{2L}{\pi}$. New moment $M' = m \times \frac{2L}{\pi} = \frac{2M}{\pi}$.

  • Full circle: The effective distance between poles is zero, so $M' = 0$.

  • Equilateral triangle shape: If bent such that poles are at two vertices of a triangle with side $L/3$, the displacement is $L/3$. New moment $M' = \frac{M}{3}$.

• Wire carrying current $I$ bent into a square of side $a$:

  • $L = 4a \rightarrow a = L/4$. $M = I \times a^2 = I (L/4)^2 = \frac{IL^2}{16}$.

• Wire carrying current $I$ bent into an equilateral triangle:

  • $L = 3a \rightarrow a = L/3$. $A = \frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{4} (L/3)^2 = \frac{\sqrt{3}L^2}{36} = \frac{L^2}{12\sqrt{3}}$. $M = \frac{IL^2}{12\sqrt{3}}$.

Magnetic Induction due to a Bar Magnet

• Axiom (Axial Point): The magnetic field is in the same direction as the magnetic moment vector ($M$).

  • $B_{axis} = \frac{\mu_0}{4\pi} \frac{2M}{r^3}$.

• Equatorial Point: The magnetic field is in the opposite direction to the magnetic moment vector ($M$).

  • $B_{eq} = \frac{\mu_0}{4\pi} \frac{M}{r^3}$.

Torque and Work Derivation

• Force on Poles: In an external field ($B$), the N-pole experiences force $F = +mB$ in the direction of the field, and the S-pole experiences $F = -mB$ opposite to the field.

• Torque Calculation: $\tau = \text{Force} \times \text{Perpendicular distance} = mB \times (2L \sin(\theta)) = (m \times 2L) B \sin(\theta) = MB \sin(\theta)$.

• Work Done ($W$):

  • $dW = \tau d\theta = MB \sin(\theta) d\theta$.

  • $W = \int_{\theta_1}^{\theta_2} MB \sin(\theta) d\theta = MB [-\cos(\theta)]_{\theta_1}^{\theta_2} = MB [\cos(\theta_1) - \cos(\theta_2)]$.