Comprehensive Study Notes on Magnetism and Matter

Properties of Magnets and Magnetic Poles

• Repulsion and Attraction: Magnetic poles exhibit specific interactive forces. There is a repulsive force when two like poles (North-North or South-South) are brought together. In contrast, an attractive force exists between unlike poles (North-South).

• Inseparability of Poles: It is impossible to isolate a single magnetic pole (a monopole). If a bar magnet is broken into two pieces, the result is two smaller, separate bar magnets, each possessing its own North and South poles, albeit with weaker magnetic properties than the original.

• Non-existence of Monopoles: Unlike electric charges, where isolated positive and negative charges can exist, magnetic monopoles have not been found in nature. All magnetic phenomena are currently explained via magnetic dipoles.

• Geographic Alignment: When a bar magnet is suspended freely, its North pole points approximately toward the Earth’s geographic north, and its South pole points toward the geographic south.

The Bar Magnet and Magnetic Field Lines

• Iron Filings Visualization: Sprinkling iron filings over a bar magnet on a glass sheet reveals a pattern of lines. This pattern suggests the magnet acts as a magnetic dipole with two distinct poles where the field is strongest.

• Magnetic Field Lines: These serve as a visual and intuitive representation of the magnetic field $B$. They share similarities with the field lines of a current-carrying solenoid.

• Core Properties of Magnetic Field Lines:

  • Continuous Loops: Magnetic field lines form continuous closed loops. This differs from electric dipole field lines, which begin at a positive charge and end at a negative charge or extend to infinity.

  • Direction: The tangent to a magnetic field line at any specific point indicates the direction of the net magnetic field $B$ at that point.

  • Magnitude: The strength of the magnetic field $B$ is proportional to the density of the field lines (the number of lines crossing a unit area).

  • Non-intersection: Magnetic field lines never intersect. If they did, the magnetic field would have two different directions at the point of intersection, which is physically impossible.

• Nomenclature Note: While some texts use the term "magnetic lines of force," this is avoided in modern physics because, unlike electrostatics, these lines do not represent the direction of force on a moving charge.

Bar Magnet as an Equivalent Solenoid

• Ampere’s Hypothesis: All magnetic phenomena are explainable through circulating currents. A bar magnet can be viewed as containing many microscopic circulating currents, analogous to a solenoid.

• Physical Analogy: Cutting a bar magnet is equivalent to cutting a solenoid; both yield two smaller entities with similar, though weaker, magnetic properties. Field lines continue to emerge from one face and enter the other.

• Axial Field Comparison: The axial magnetic field of a finite solenoid at a far distance $r$ (where $r$ is much larger than the dimensions of the solenoid) is given by:     $B = \frac{\mu_0}{4\pi} \frac{2m}{r^3}$

• Magnetic Moment Match: The magnetic moment $m$ of a bar magnet is equal to the magnetic moment of an equivalent solenoid that produces an identical magnetic field.

Dipole in a Uniform Magnetic Field

• Torque on a Needle: When a magnetic needle with magnetic moment $m$ is placed in a uniform magnetic field $B$, it experiences a torque $\tau$ defined as:     $\tau = \mathbf{m} \times \mathbf{B}$     The magnitude of the restoring torque is:     $\tau = mB \sin(\theta)$     where $\theta$ is the angle between the magnetic moment $\mathbf{m}$ and the magnetic field $\mathbf{B}$.

• Magnetic Potential Energy: The potential energy $U_m$ is calculated by integrating the torque over the angle:     $U_m = \int \tau(\theta) d\theta = \int mB \sin(\theta) d\theta = -mB \cos(\theta)$     In vector form:     $U_m = -\mathbf{m} \cdot \mathbf{B}$

• Stability Positions:

  • Most Stable: At $\theta = 0^{\circ}$, $U_m = -mB$ (minimum potential energy).

  • Most Unstable: At $\theta = 180^{\circ}$, $U_m = +mB$ (maximum potential energy).

  • Zero Reference: By convention, the zero of potential energy is often set at $\theta = 90^{\circ}$.

The Electrostatic Analog

• Mathematical Correspondences: Magnetic field equations mirror electric field equations through specific replacements:

  • $E \rightarrow B$

  • $p \rightarrow m$

  • $\frac{1}{\epsilon_0} \rightarrow \mu_0$

• Comparison Equations (for distance $r \gg l$):

  • Equatorial Field: $B_E = -\frac{\mu_0}{4\pi} \frac{m}{r^3}$

  • Axial Field: $B_A = \frac{\mu_0}{4\pi} \frac{2m}{r^3}$

• External Field Comparison:

  • Torque: Electric field is $\mathbf{p} \times \mathbf{E}$; Magnetic field is $\mathbf{m} \times \mathbf{B}$.

  • Energy: Electric field is $-\mathbf{p} \cdot \mathbf{E}$; Magnetic field is $-\mathbf{m} \cdot \mathbf{B}$.

Magnetism and Gauss’s Law

• Definition: Gauss’s law for magnetism states that the net magnetic flux $\phi_B$ through any closed surface $S$ is zero:     $\phi_B = \sum_{all} \mathbf{B} \cdot \Delta\mathbf{S} = 0$

• Physical Significance: This law reflects the absence of magnetic monopoles. Every field line entering a closed surface must also exit it, as magnetic field lines form closed loops. There are no sources or sinks of the magnetic field $B$.

• Comparison with Electrostatics: In electrostatics, the flux through a closed surface equals $\frac{q}{\epsilon_0}$. Because there are no isolated magnetic charges (monopoles), the magnetic equivalent of $q$ is always zero for a closed volume.

Magnetisation and Magnetic Intensity

• Magnetisation ($M$): Defined as the net magnetic moment per unit volume of a material:     $\mathbf{M} = \frac{\mathbf{m}_{net}}{V}$     Its dimensions are $L^{-1} A$ and units are $\text{A m}^{-1}$.

• Magnetic Intensity ($H$): A vector field related to external currents (like those in a solenoid) that influence a material. It is defined as:     $\mathbf{H} = \frac{\mathbf{B}}{\mu_0} - \mathbf{M}$

• Total Magnetic Field ($B$): The total field inside a material is the sum of the field due to external currents and the field due to the material's magnetisation:     $\mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M})$

• Magnetic Susceptibility ($\chi$): A dimensionless quantity measuring how a material responds to an external field:     $\mathbf{M} = \chi \mathbf{H}$

• Relative Magnetic Permeability ($\mu_r$): Defined as:     $\mu_r = 1 + \chi$

• Magnetic Permeability ($\mu$): The permeability of the substance is:     $\mu = \mu_0 \mu_r = \mu_0(1 + \chi)$     The total field can thus be rewritten as: $\mathbf{B} = \mu_0 \mu_r \mathbf{H} = \mu \mathbf{H}$.

Classification of Magnetic Materials

• Diamagnetism:

  • Susceptibility: $\chi$ is small and negative (-1 \le \chi < 0).

  • Permeability: 0 \le \mu_r < 1.

  • Behavior: Diamagnetic substances are repelled by magnets and move from stronger to weaker parts of a non-uniform field. Field lines are expelled from the material.

  • Mechanism: Induced magnetic moments are created in the opposite direction of the applied field (Lenz's Law effect on orbiting electrons).

  • Superconductors: Exhibit perfect diamagnetism ($\chi = -1, \mu_r = 0$), a phenomenon called the Meissner effect.

  • Examples: Bismuth, copper, lead, silicon, water, nitrogen (STP), sodium chloride.

• Paramagnetism:

  • Susceptibility: $\chi$ is small and positive (0 < \chi < \epsilon).

  • Permeability: 1 < \mu_r < 1 + \epsilon.

  • Behavior: Paramagnetic substances are weakly attracted to magnets and move from weaker to stronger parts of a field. Field lines are slightly concentrated within the material.

  • Mechanism: Atoms possess permanent magnetic moments, but thermal motion causes random orientation. External fields align these dipoles.

  • Examples: Aluminum, sodium, calcium, oxygen (STP), copper chloride.

• Ferromagnetism:

  • Susceptibility: $\chi$ is large and positive ($\chi \gg 1$).

  • Permeability: $\mu_r \gg 1$.

  • Behavior: Strongly attracted to magnets. Field lines are highly concentrated.

  • Domains: Microscopic regions (size $\sim 1\text{ mm}$, containing $\sim 10^{11}$ atoms) where moments are spontaneously aligned.

  • Hard Ferromagnets: Retain magnetisation after the external field is removed (e.g., Alnico, lodestone). Used for permanent magnets.

  • Soft Ferromagnets: Disappearance of magnetisation when the external field is removed (e.g., soft iron).

  • Temperature Dependence: Ferromagnetic properties disappear at high temperatures as domains disintegrate; the material then becomes paramagnetic.

  • Examples: Iron, cobalt, nickel, gadolinium, alnico.

Questions & Discussion

• Question: What happens if a bar magnet is cut transverse to its length or along its length?

  • Answer: In both cases, the result is two separate magnets, each having its own North and South poles.

• Question: Why does an iron nail near a bar magnet experience an attractive force in addition to torque, whereas a needle in a uniform field only feels torque?

  • Answer: In a uniform field, forces on the poles cancel out, leaving only torque. Near a bar magnet, the field is non-uniform. The nail develops an induced dipole moment; because the induced opposite pole is closer to the magnet than the like pole, a net attractive force occurs.

• Question: Must every magnetic configuration have a N and S pole? What about a toroid?

  • Answer: Not necessarily. A configuration only has poles if it has a net non-zero magnetic moment. A toroid or an infinite straight conductor has no set poles.

• Question: How can one distinguish between two identical iron bars, A and B, if only one is magnetised?

  • Answer: Pick up bar A and touch its end to the middle of bar B. If bar A experiences no force at the center of B, then B is the magnet (as the field is weakest at the center of a bar magnet). If the force remains constant from end to middle, A is the magnet.

• Question: Why is it misleading to call magnetic field lines "lines of force"?

  • Answer: The magnetic force on a moving charge is always perpendicular to the magnetic field $\mathbf{B}$ ($F = q\mathbf{v} \times \mathbf{B}$), not along the field lines.

• Question: Can a system have a magnetic moment with zero net charge?

  • Answer: Yes. For example, in paramagnetic materials, atoms have zero net charge but possess net magnetic moments due to the motion of electrons in current loops.

EX 5.1

(a) Cutting a Bar Magnet
When a bar magnet is cut:

• Transversely: Each piece becomes a smaller magnet with a north and south pole.

• Along its Length: Similarly, both pieces will still have their poles, resulting in two smaller magnets.

(b) Magnetized Needle in a Uniform Field
A magnetized needle in a uniform magnetic field experiences torque (a twisting force) but not net force, because it is aligned with the field; thus, no overall push or pull occurs. In contrast, an iron nail near a bar magnet experiences attraction, as it becomes magnetized by the bar magnet and the induced poles create a net attractive force.

(c) Magnetic Configuration
Every magnetic object typically has both a north and south pole. However, theoretical configurations like toroids (looped wires) might not have distinct poles if there’s no net magnetic moment.

(d) Identifying Magnetization
To determine if two identical rods are magnetized, one could move their ends closer together to check for attraction. The magnetized one will induce a magnetic moment in the non-magnetized one, showing attraction or repulsion. If there’s activity observed when bringing the ends together, the one exhibiting force is magnetized.