Magnetic Properties of Matter: HRK Physics Chapter 37 Complete Notes & Formulas

Complete Guide to Gauss's Law for Magnetism, Magnetic Dipole Moments, Bohr Magneton, Paramagnetism & Diamagnetism - B.Sc. Physics Edition

Magnetic Properties HRK Chapter 37 Gauss's Law for Magnetism Magnetic Dipole Moment Bohr Magneton Reading Time: 25 min

📋 Table of Contents

• 1. Gauss's Law for Magnetism
• 2. Magnetic Dipole Moment of Orbiting Electron
  • 2.1 Derivation of Orbital Magnetic Moment
  • 2.2 Bohr Magneton
• 3. Intrinsic Magnetic Moments
• 4. Total Magnetic Moment of an Atom
• 5. Nuclear Magnetism
• 6. Magnetization
• 7. Magnetic Materials
  • 7.1 Paramagnetism
  • 7.2 Diamagnetism
• 8. Problem Solving
• Frequently Asked Questions

📜 Historical Background

The study of magnetic properties of matter dates back to ancient times, with the discovery of naturally occurring magnetic minerals like magnetite. However, the scientific understanding of magnetism developed significantly in the 19th and 20th centuries.

Key developments include:

• Hans Christian Ørsted's discovery of electromagnetism (1820)
• Michael Faraday's work on diamagnetism and paramagnetism (1845)
• Pierre Curie's investigation of magnetic susceptibility and temperature (1895)
• The development of quantum mechanics explaining atomic magnetic moments

These discoveries laid the foundation for our modern understanding of magnetic materials and their applications.

Gauss's Law for Magnetism

🔬 Fundamental Difference Between Electric and Magnetic Dipoles

An electric dipole consists of two equal but opposite point charges separated by a small distance. In contrast, a magnetic dipole is considered as a combination of a north pole and a south pole separated by a small distance.

The crucial difference is that electric dipoles can be separated into individual charges, but magnetic dipoles cannot be isolated into separate poles. Each attempt to divide a magnetic dipole results in new pairs of poles.

📝 Gauss's Law for Magnetic Fields

For electric fields, Gauss's law states:

\[ \Phi_E = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{q}{\varepsilon_0} \]

For magnetic fields, since there are no isolated magnetic charges (magnetic monopoles), the net magnetic flux through any closed surface is always zero:

\[ \Phi_B = \oint \mathbf{B} \cdot d\mathbf{A} = 0 \]

This mathematical statement reflects the experimental observation that magnetic field lines always form closed loops, with no beginning or end.

💡 Key Insight: No Magnetic Monopoles

The absence of magnetic monopoles is a fundamental property of nature. Every time we try to separate a magnetic dipole, we create new pairs of north and south poles. This property holds true even at the microscopic level of individual atoms, where each atom behaves as a magnetic dipole with both north and south poles.

Magnetic Dipole Moment of Orbiting Electron

🔍 Atomic Origin of Magnetism

Electron Orbital Motion

• Electron with charge -e

• Circular orbit radius r

• Orbital speed v

• Equivalent to current loop

The orbiting electron creates a microscopic current loop with associated magnetic moment

Key Observations

• Moving electrons in atoms form microscopic current loops
• Each current loop has an associated magnetic dipole moment
• The magnetic moment is proportional to angular momentum
• Vectors μ and l point in opposite directions due to negative charge

Derivation of Orbital Magnetic Moment

🧮 Mathematical Derivation

Step 1: Current in the Electron Orbit

The orbiting electron is equivalent to a current loop with current:

\[ I = \frac{-e}{T} \]

where T is the orbital period.

Step 2: Orbital Period

The orbital period can be expressed as:

\[ T = \frac{2\pi r}{v} \]

Step 3: Magnetic Dipole Moment

The magnetic dipole moment for a current loop is:

\[ \mu_l = I \cdot A \]

\[ = \left( -\frac{e}{T} \right) \cdot (\pi r^2) \]

\[ = -\frac{e}{T} \cdot \pi r^2 \]

Step 4: Substituting for T

\[ \mu_l = -\frac{e}{(2\pi r/v)} \cdot \pi r^2 \]

\[ = -\frac{ev}{2\pi r} \cdot \pi r^2 \]

\[ = -\frac{evr}{2} \]

Step 5: Relation to Angular Momentum

Multiplying numerator and denominator by electron mass m_e:

\[ \mu_l = -\frac{e}{2m_e} \cdot (m_e v r) \]

Since the orbital angular momentum l = m_evr:

\[ \mu_l = -\left( \frac{e}{2m_e} \right) l \]

Step 6: Vector Form

In vector form, for all electrons in an atom:

\[ \mu_L = -\left( \frac{e}{2m_e} \right) L \]

where L = Σl is the total orbital angular momentum.

Bohr Magneton

Quantum Mechanical Quantization

From quantum mechanics, angular momentum is quantized in units of ħ = h/2π:

\[ L = n\hbar = n\frac{h}{2\pi} \]

Definition of Bohr Magneton

The Bohr magneton μ_B is defined as the magnetic moment when L = ħ:

\[ \mu_B = \frac{e}{2m_e} \cdot \frac{h}{2\pi} \]

\[ = \frac{eh}{4\pi m_e} \]

Numerical Value

Substituting values for e, h, and m_e:

\[ \mu_B = 9.27 \times 10^{-24} \, \text{J/T} \]

Physical Significance

The Bohr magneton represents the fundamental quantum of magnetic moment associated with electron orbital motion. Magnetic moments in atoms are typically on the order of μ_B.

Intrinsic Magnetic Moments

📜 Discovery of Electron Spin

Experiments in the 1920s, involving atomic beams passing through magnetic fields, revealed that the orbital magnetic moment alone couldn't explain observed magnetic properties. This led to the discovery of intrinsic magnetic moment associated with electron spin.

🔍 Spin Magnetic Moment

For a single electron, the relationship between intrinsic angular momentum s and intrinsic magnetic moment μ_s is:

\[ \mu_s = -\left( \frac{e}{m_e} \right) s \]

Note the factor of 2 difference compared to the orbital magnetic moment formula. This arises because the basic unit of spin angular momentum is ħ/2, while for orbital angular momentum it's ħ.

💡 Fundamental Difference

Orbital angular momentum and orbital magnetic moments are properties of an electron's particular state of motion. In contrast, intrinsic (spin) angular momentum and intrinsic magnetic moments are fundamental characteristics of the electron itself, along with its mass and charge.

Nuclear Magnetism

🧬 Nuclear Composition

Atomic nuclei consist of protons and neutrons in orbital motion under mutual forces. Each nucleon also has intrinsic angular momentum, contributing to nuclear magnetic moments.

📏 Nuclear Magneton

The nuclear magneton μ_N is defined similarly to the Bohr magneton but using the proton mass m_p:

\[ \mu_N = \frac{eh}{4\pi m_p} \]

Since m_p ≈ 1836m_e, μ_N ≈ μ_B/1836.

⚡ Magnetic Resonance

Nuclear magnetic moments are the basis for nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI), powerful techniques for studying molecular structure and medical imaging.

Magnetic Materials

📊 Classification of Magnetic Materials

Materials can be classified based on their response to external magnetic fields:

• Diamagnetic: Weakly repelled by magnetic fields
• Paramagnetic: Weakly attracted by magnetic fields
• Ferromagnetic: Strongly attracted by magnetic fields
• Antiferromagnetic: Special ordered state with zero net magnetization
• Ferrimagnetic: Similar to ferromagnetic but with different sublattice magnetizations

Paramagnetism

🔬 Paramagnetic Materials

Paramagnetic materials contain atoms or ions with permanent magnetic dipole moments that tend to align with an external magnetic field. In the absence of an external field, these moments are randomly oriented due to thermal agitation.

📝 Key Characteristics

• Positive magnetic susceptibility (χ > 0)
• Weak attraction to magnetic fields
• Magnetization disappears when external field is removed
• Follows Curie's law: χ ∝ 1/T
• Examples: Aluminum, platinum, oxygen

Diamagnetism

🔬 Diamagnetic Materials

Diamagnetic materials have no permanent magnetic dipole moments. When placed in an external magnetic field, induced currents in atomic electron orbits create magnetic moments that oppose the applied field.

📝 Key Characteristics

• Negative magnetic susceptibility (χ < 0)
• Weak repulsion from magnetic fields
• Present in all materials but often masked by stronger effects
• Independent of temperature
• Examples: Copper, silver, gold, bismuth

Property	Paramagnetism	Diamagnetism
Origin	Permanent atomic dipoles	Induced atomic currents
Susceptibility (χ)	Positive, small (10-5 to 10-3)	Negative, very small (≈ -10-5)
Temperature Dependence	χ ∝ 1/T (Curie's Law)	Independent of temperature
Response to Field	Weak attraction	Weak repulsion
Examples	Al, Pt, O2	Cu, Ag, Bi, H2O

Problem Solving Approach

🎯 Step-by-Step Methodology

Sample Problem:

Calculate the orbital magnetic moment of an electron in the first Bohr orbit of a hydrogen atom. The radius of the orbit is 5.29 × 10^-11 m and the orbital speed is 2.19 × 10⁶ m/s.

Step 1: Identify Known Values

Electron charge: e = 1.6 × 10^-19 C

Orbit radius: r = 5.29 × 10^-11 m

Orbital speed: v = 2.19 × 10⁶ m/s

Step 2: Calculate Equivalent Current

The orbiting electron is equivalent to a current loop with:

\[ I = \frac{-e}{T} = \frac{-ev}{2\pi r} \]

\[ = \frac{-(1.6 \times 10^{-19}) \cdot (2.19 \times 10^6)}{2\pi \cdot (5.29 \times 10^{-11})} \]

\[ = -1.05 \times 10^{-3} \, \text{A} \]

Step 3: Calculate Area of the Orbit

\[ A = \pi r^2 = \pi \cdot (5.29 \times 10^{-11})^2 \]

\[ = 8.79 \times 10^{-21} \, \text{m}^2 \]

Step 4: Calculate Magnetic Moment

\[ \mu_l = I \cdot A \]

\[ = (-1.05 \times 10^{-3}) \cdot (8.79 \times 10^{-21}) \]

\[ = -9.23 \times 10^{-24} \, \text{J/T} \]

Step 5: Compare with Bohr Magneton

The magnitude is very close to the Bohr magneton (9.27 × 10^-24 J/T), confirming our calculation.

Frequently Asked Questions

❓ Why is the spin magnetic moment of an electron exactly twice the orbital magnetic moment for the same angular momentum?

This is a fundamental result from quantum electrodynamics (QED), the quantum theory of electromagnetic interactions. The factor of 2, known as the electron g-factor, arises from relativistic effects and the quantum nature of the electron. For orbital motion, g = 1, while for spin, g ≈ 2 (actually 2.002319...). This difference reflects the fact that orbital angular momentum is associated with the electron's motion through space, while spin is an intrinsic property.

❓ If magnetic monopoles don't exist, why do we still search for them?

While Gauss's law for magnetism states that no magnetic monopoles have been observed, many theoretical frameworks (including grand unified theories) predict their existence. Finding magnetic monopoles would:

• Explain the quantization of electric charge
• Provide symmetry between electricity and magnetism
• Validate certain unified field theories
• Revolutionize our understanding of fundamental physics

Despite extensive searches, no confirmed detection has been made to date.

❓ What's the practical significance of the Bohr magneton?

Application	Significance
Atomic Physics	Fundamental unit for atomic magnetic moments
Magnetic Resonance	Determines energy level splitting in magnetic fields
Material Science	Used to characterize magnetic properties of materials
Quantum Computing	Important for spin-based quantum bits (qubits)

📚 Master Electromagnetism

Understanding magnetic properties of matter is essential for advanced studies in condensed matter physics, materials science, and modern technology development.

Read More: Physics HRK Notes of Electricity and Magnetism

© 2025 Physics Education Initiative | HRK Physics Chapter 37: Magnetic Properties of Matter

Based on Halliday, Resnick, and Krane's "Physics" with additional insights from university physics curriculum

Author: Muhammad Ali Malik | Physics Educator | Contact: +923016775811