Physics: Class 12 (Book 1) — Chapter 5: Magnetism and Matter

• The earth behaves as a giant magnetic dipole, with its magnetic field pointing approximately from the geographic south to the geographic north.
• When a bar magnet is freely suspended, it aligns itself along the geographic north-south direction. The tip pointing north is designated the north pole , and the tip pointing south is the south pole .
• Like magnetic poles repel each other, whereas unlike magnetic poles attract each other.
• Isolated magnetic poles ( magnetic monopoles ) do not exist. Slicing a bar magnet in half yields two smaller, complete bar magnets, each possessing its own north and south poles.
• It is possible to artificially magnetise iron and its various alloys.

Magnetic field lines are visual representations used to map the strength and direction of a magnetic field. Their fundamental properties include:

1. Continuous Closed Loops: Unlike electric field lines which begin on positive charges and terminate on negative charges, magnetic field lines form continuous closed loops, running from the north pole to the south pole outside the magnet, and from the south pole to the north pole inside.
2. Direction of Field Vector: The tangent drawn to a magnetic field line at any given point represents the direction of the net magnetic field vector $\vec{B}$ at that point.
3. Field Density: The number of field lines crossing a unit area perpendicular to the field is proportional to the magnitude of the magnetic field $\vec{B}$. The field is stronger near the poles where the lines are highly concentrated.
4. Non-intersection: Magnetic field lines never intersect. If they did, the magnetic field vector would point in two different directions at the intersection point, which is physically impossible.

A bar magnet can be modeled as a large number of circulating atomic currents, analogous to a finite solenoid. The axial magnetic field of a finite solenoid of length $2l$, radius $a$, and having $n$ turns per unit length carrying current $I$, at a far axial distance $r$ ($r \gg a$ and $r \gg l$), is given by:

$$B = \frac{\mu_0}{4\pi} \frac{2m}{r^3}$$

where $m = (N I) A = (n \cdot 2l) I (\pi a^2)$ is the total magnetic moment of the solenoid. This expression is mathematically identical to the far axial magnetic field of a bar magnet, establishing their physical equivalence.

When a magnetic dipole of magnetic moment $\vec{m}$ is placed in a uniform magnetic field $\vec{B}$ at an angle $\theta$, it experiences a restoring torque $\vec{\tau}$:

$$\vec{\tau} = \vec{m} \times \vec{B}$$

The magnitude of this torque is:

$$\tau = mB \sin\theta$$

The work done by an external agent in rotating the dipole is stored as magnetic potential energy $U_m$:

$$U_m = \int \tau(\theta) d\theta = \int mB \sin\theta d\theta = -mB \cos\theta = -\vec{m} \cdot \vec{B}$$

• Stable Equilibrium: Occurs at $\theta = 0^\circ$, where $U_m = -mB$ (minimum potential energy).
• Unstable Equilibrium: Occurs at $\theta = 180^\circ$, where $U_m = +mB$ (maximum potential energy).
• Zero Reference: Typically chosen at $\theta = 90^\circ$, where $U_m = 0$.

Mathematical Problem Statement: A short bar magnet placed with its axis at 30° with a uniform external magnetic field of 0.25 T experiences a torque of magnitude equal to $4.5 \times 10^{-2}\text{ J}$. What is the magnitude of the magnetic moment of the magnet?

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Step-by-Step Algorithmic Solution Trace:

1. Identify the given parameters: torque $\tau = 4.5 \times 10^{-2}\text{ J}$, magnetic field $B = 0.25\text{ T}$, and angle $\theta = 30^\circ$.
2. Use the torque formula: $\tau = mB \sin\theta$.
3. Isolate the magnetic moment $m$: $$m = \frac{\tau}{B \sin\theta}$$
4. Substitute the values into the equation: $$m = \frac{4.5 \times 10^{-2}}{0.25 \times \sin(30^\circ)} = \frac{4.5 \times 10^{-2}}{0.25 \times 0.5} = \frac{4.5 \times 10^{-2}}{0.125}$$
5. Calculate the final value: $m = 0.36\text{ J T}^{-1}$.

Final Evaluation Output: $m = \mathbf{0.36\text{ J T}^{-1}}$

The mathematical formulations for magnetic dipoles are directly analogous to those of electric dipoles under specific variable transformations:

• Magnetisation ($\vec{M}$): Defined as the net magnetic dipole moment per unit volume of the material: $$\vec{M} = \frac{\vec{m}_{\text{net}}}{V}$$ Units: $\text{A m}^{-1}$, Dimensions: $[L^{-1} A]$.
• Magnetic Intensity ($\vec{H}$): Represents the external magnetic field contribution, defined as: $$\vec{H} = \frac{\vec{B}}{\mu_0} - \vec{M}$$ Units: $\text{A m}^{-1}$.
• Total Magnetic Field ($\vec{B}$): The total magnetic field inside a material is the vector sum of the field due to external currents ($\mu_0 \vec{H}$) and the field due to the material's magnetisation ($\mu_0 \vec{M}$): $$\vec{B} = \mu_0 (\vec{H} + \vec{M})$$
• Magnetic Susceptibility ($\chi$): A dimensionless parameter measuring how easily a material magnetises in response to an external field: $$\vec{M} = \chi \vec{H}$$
• Permeability Relations: $$\vec{B} = \mu_0 (1 + \chi) \vec{H} = \mu_0 \mu_r \vec{H} = \mu \vec{H}$$ where $\mu_r = 1 + \chi$ is the relative magnetic permeability , and $\mu = \mu_0 \mu_r$ is the absolute magnetic permeability of the medium.

Diamagnetic substances are repelled by magnets. When placed in an external magnetic field, the field lines are expelled from the material, reducing the internal magnetic field.

• Mechanism: Orbiting electrons speed up or slow down to induce a magnetic moment opposite to the applied field (in accordance with Lenz's law).
• Examples: Bismuth, copper, lead, silicon, nitrogen (at STP), water, and sodium chloride.
• Superconductors: Exhibit perfect diamagnetism ($\chi = -1$, $\mu_r = 0$). They completely expel magnetic field lines, a phenomenon known as the Meissner effect .

Paramagnetic substances get weakly magnetised in the direction of the external magnetic field. Field lines are concentrated inside the material.

• Mechanism: Individual atoms possess permanent magnetic dipole moments, which align with the external field against the randomising effect of thermal motion.
• Examples: Aluminium, sodium, calcium, oxygen (at STP), and copper chloride.

Ferromagnetic substances get strongly magnetised in an external magnetic field and are strongly attracted to magnets.

• Mechanism: Atomic dipoles spontaneously align in macroscopic regions called domains (typically 1 mm in size containing $\sim 10^{11}$ atoms). Under an external field, these domains grow and align collectively.
• Hard Ferromagnets: Retain magnetisation even after the external field is removed (e.g., Alnico, lodestone). Used to make permanent magnets.
• Soft Ferromagnets: Lose magnetisation quickly when the external field is removed (e.g., soft iron). Used in electromagnets and transformer cores.
• Temperature Dependence: At high temperatures, thermal agitation disintegrates the domain structure, and a ferromagnet transitions into a paramagnet.