Unit 3: Magnetism and Magnetic Effects of Electric Current Study Notes

Introduction to Magnetism

• Historical Origins: The word 'magnetism' is derived from iron ore magnetite ($Fe_3O_4$). Historically, magnets were used for navigation (compass), magnetic therapy, and magic shows.
• Modern Applications: Magnets are found in motors, cycle dynamos, loudspeakers, magnetic tapes (audio/video), mobile phones, headphones, CDs, pen-drives, hard discs, refrigerators, and generators.
• Unification of Electricity and Magnetism: Initially thought to be independent branches, electricity and magnetism were unified into 'electromagnetism' in 1820 when H.C. Oersted observed a current-carrying wire deflecting a magnetic compass needle.
• Biological Magnetism: Many birds and animals possess a magnetic sense. For example, Zebrafinches use Earth's magnetic field for navigation due to the protein cryptochrome Cry4 present in their retina.
• Magnetic Levitation: Examples include the Maglev train, which uses sets of magnets to float above a guideway and achieve great speeds with low friction.

Earth’s Magnetic Field and Magnetic Elements

• Theories of Earth's Magnetism:     * William Gilbert (1600): Proposed Earth behaves like a gigantic bar magnet. However, high internal temperatures make it impossible for materials to retain magnetism.     * Gover's Theory: Suggested hot rays from the sun heat air at the equator; this air electrifies as it moves toward the poles, magnetizing ferromagnetic materials on the surface.
• Geomagnetism / Terrestrial Magnetism: The study of Earth’s magnetic field. The Earth's magnetic south pole is near the geographic north pole, and its magnetic north pole is near the geographic south pole.
• Geographic and Magnetic Terms:     * Geographic Axis: The axis around which Earth spins.     * Geographic Meridian: A vertical plane passing through the geographic axis.     * Geographic Equator: A great circle perpendicular to the geographic axis.     * Magnetic Axis: The line connecting the Earth's magnetic poles.     * Magnetic Meridian: A vertical plane passing through the magnetic axis.     * Magnetic Equator: A great circle perpendicular to the magnetic axis.
• The Three Elements of Earth's Magnetic Field:     1. Magnetic Declination (D): The angle between the magnetic meridian and the geographic meridian at a point. It is higher at higher latitudes and smaller near the equator. (For Chennai: $-1^{\circ} \, 8'$ or west).     2. Magnetic Dip or Inclination (I): The angle subtended by the Earth's total magnetic field $\mathbf{B_E}$ with the horizontal direction in the magnetic meridian. (For Chennai: $14^{\circ} \, 16'$).     3. Horizontal Component ($B_H$): The component of Earth’s magnetic field along the horizontal direction in the magnetic meridian.
• Mathematical Relationships:     * Horizontal component: $B_H = B_E \cos(I)$     * Vertical component: $B_V = B_E \sin(I)$     * Relationship: $\tan(I) = \frac{B_V}{B_H}$
• Specific Locations:     * At the Magnetic Equator: $I = 0^{\circ}$, so $B_H = B_E$ (maximum) and $B_V = 0$.     * At the Magnetic Poles: $I = 90^{\circ}$, so $B_H = 0$ and $B_V = B_E$ (maximum).
• Aurora Borealis and Aurora Australis: Natural light displays at high latitudes caused by the interaction of atmospheric gaseous particles with solar wind charged particles guided by Earth's magnetic field. Oxygen collisions produce pale yellowish-green; nitrogen produces blue or purplish-red.

Basic Properties of Magnets

• Magnetic Dipole Moment ($p_m$): Defined as the product of pole strength ($q_m$) and magnetic length ($d = 2l$). It is a vector quantity directed from the south pole to the north pole.     * Formula: $\mathbf{p_m} = q_m \mathbf{d}$     * Magnitude: $p_m = 2 q_m l$     * SI Unit: $\text{A m}^2$
• Magnetic Field ($B$): The region where a unit pole strength experiences a force.     * Formula: $\mathbf{B} = \frac{\mathbf{F}}{q_m}$     * SI Unit: $\text{N A}^{-1} \, \text{m}^{-1}$
• Types of Magnets:     * Natural Magnets: Irregular shapes, weak strength (e.g., Magnetite).     * Artificial Magnets: Man-made with specific shapes (e.g., bar, cylindrical).
• Key Properties:     1. Freely suspended magnets point North-South.     2. Magnets attract magnetic substances; force is maximum at the ends.     3. Breaking a magnet results in two new magnets (no monopoles).     4. Both poles have equal strength ($q_m$).     5. Magnetic length vs. Geometrical length: Magnetic length is the distance between poles; Geometrical length is the actual physical length. The ratio is approximately $\frac{5}{6} \approx 0.833$.
• Magnetic Field Lines:     * Continuous closed curves.     * Direction: North to South outside; South to North inside.     * Tangents at any point show the magnetic field direction.     * Lines never intersect.     * Closeness of lines indicates field strength.
• Magnetic Flux ($\Phi_B$): The number of field lines crossing a unit area.     * Formula: $\Phi_B = \mathbf{B} \cdot \mathbf{A} = B A \cos(\theta)$     * Scalar quantity. SI Unit: Weber ($\text{Wb}$). CGS Unit: Maxwell ($1 \text{ Wb} = 10^8 \text{ maxwell}$).     * Dimensional formula: $M L^2 T^{-2} A^{-1}$.
• Uniform and Non-Uniform Fields:     * Uniform: Same magnitude and direction everywhere (e.g., Earth's field locally).     * Non-Uniform: Magnitude or direction varies (e.g., bar magnet field).

Coulomb’s Inverse Square Law of Magnetism

• Statement: The force of attraction or repulsion between two magnetic poles is directly proportional to the product of their pole strengths and inversely proportional to the square of the distance between them.
• Mathematical Form: $F = k \frac{q_{mA} q_{mB}}{r^2}$     * In SI units, $k = \frac{\mu_0}{4 \pi} = 10^{-7} \, \text{H m}^{-1}$.     * $\mu_0$ is the permeability of free space.

Magnetic Field Calculations for a Bar Magnet

• Point on Axial Line:     * Distance $r$ from the center O.     * Formula for a short magnet ($r \gg l$): $\mathbf{B_{axial}} = \frac{\mu_0}{4\pi} \frac{2\mathbf{p_m}}{r^3}$     * Direction: Along the magnetic dipole moment (South to North).
• Point on Equatorial Line:     * Distance $r$ from the center O on the perpendicular bisector.     * Formula for a short magnet ($r \gg l$): $\mathbf{B_{equatorial}} = -\frac{\mu_0}{4\pi} \frac{\mathbf{p_m}}{r^3}$     * Direction: Opposite to the magnetic dipole moment (North to South).
• Ratio: The magnitude of $B_{axial}$ is twice that of $B_{equatorial}$ at the same distance.

Torque and Potential Energy

• Torque ($\tau$): When a bar magnet is placed in a uniform field $\mathbf{B}$ at an angle $\theta$, it experiences a couple.     * Formula: $\mathbf{\tau} = \mathbf{p_m} \times \mathbf{B}$     * Magnitude: $\tau = p_m B \sin(\theta)$     * Net force is zero in a uniform field; however, in a non-uniform field, both torque and translatory force exist.
• Time Period of Oscillation ($T$): For a magnet oscillating in a uniform field:     * Formula: $T = 2\pi \sqrt{\frac{I}{p_m B_H}}$     * Where $I$ is the moment of inertia.
• Potential Energy ($U$): Work done in rotating a dipole from $\theta'$ to $\theta$.     * Formula: $U = -p_m B \cos(\theta) = -\mathbf{p_m} \cdot \mathbf{B}$     * Minimum Energy ($-p_m B$): At $\theta = 0^{\circ}$ (Stable equilibrium).     * Maximum Energy ($+p_m B$): At $\theta = 180^{\circ}$ (Unstable equilibrium).

Tangent Law and Tangent Galvanometer

• Tangent Law: When a magnetic needle is suspended in two mutually perpendicular uniform fields ($B$ and $B_H$), it comes to rest at an angle $\theta$ such that $B = B_H \tan(\theta)$.
• Tangent Galvanometer (TG): A device used to measure small currents.     * Construction: Circular copper coil (2, 5, or 50 turns) on a non-magnetic frame. A compass box is placed at the center.     * Magnetic Field at Center: $B = \frac{\mu_0 N I}{2 R}$     * Current Formula: $I = \left( \frac{2 R B_H}{\mu_0 N} \right) \tan(\theta) = K \tan(\theta)$     * $K$ is the reduction factor. TG is most sensitive at $\theta = 45^{\circ}$.

Magnetic Properties of Materials

• Magnetising Field ($H$): The field used to magnetize a specimen. SI Unit: $\text{A m}^{-1}$.
• Magnetic Permeability ($\mu$): The ability of a material to allow field lines to pass through.     * Relative Permeability: $\mu_r = \frac{\mu}{\mu_0}$. ($\mu_r = 1$ for vacuum).
• Intensity of Magnetisation ($M$): Net magnetic moment per unit volume.     * Formula: $M = \frac{p_m}{V} = \frac{q_m}{A}$
• Magnetic Induction (Total Field $B$): Sum of field in vacuum and field due to induced magnetisation.     * Formula: $\mathbf{B} = \mu_0 (\mathbf{H} + \mathbf{M}) = \mu_0 \mu_r \mathbf{H}$
• Magnetic Susceptibility ($\chi_m$): Measures how easily a material can be magnetized.     * Formula: $\chi_m = \frac{M}{H}$     * Relation: $\mu_r = 1 + \chi_m$

Classification of Magnetic Materials

1. Diamagnetic Materials: Atoms have no net magnetic moment. In an external field, they develop induced moments opposite to the field.     * Properties: $\chi_m$ is negative; $\mu_r < 1$; repelled by magnets; independent of temperature.     * Examples: Bismuth, Copper, Water, Gold.     * Superconductors: Perfect diamagnets ($\chi_m = -1$). The expulsion of flux is the Meissner Effect.
2. Paramagnetic Materials: Atoms possess permanent moments, but are randomly oriented. External fields align them.     * Properties: $\chi_m$ is positive and small; $\mu_r > 1$; attracted to stronger fields.     * Curie's Law: $\chi_m \propto \frac{1}{T}$     * Examples: Aluminium, Platinum, Oxygen.
3. Ferromagnetic Materials: Spontaneous alignment of moments in small regions called Domains.     * Properties: $\chi_m$ is large and positive; $\mu_r$ is very large; strong attraction.     * Curie-Weiss Law: Above Curie Temperature ($T_c$), they become paramagnetic ($\chi_m = \frac{C}{T - T_c}$).     * Examples: Iron, Nickel, Cobalt.

Hysteresis

• Definition: The phenomenon of the magnetic induction ($B$) lagging behind the magnetising field ($H$) during a cycle of magnetisation.
• Definitions in the Hysteresis Loop:     * Retentivity (Remanence): The residual magnetism left in the material when $H = 0$.     * Coercivity: The reverse magnetising field required to make residual magnetism zero.
• Hysteresis Loss: Energy lost as heat during a magnetisation cycle, equal to the area of the hysteresis loop.
• Hard vs. Soft Magnetic Materials:     * Soft (e.g., Soft Iron): Small area, low retentivity, low coercivity. Used in transformer cores and electromagnets.     * Hard (e.g., Steel, Alnico): Large area, high retentivity, high coercivity. Used for permanent magnets.

Magnetic Effects of Current

• Oersted Experiment: Observed that current deflects a compass needle, indicating current creates a magnetic field.
• Rules for Direction:     * Right Hand Thumb Rule: Thumb in direction of current; curled fingers show field lines (concentric circles for straight wires).     * Maxwell’s Right Hand Cork Screw Rule: Direction of screw advancement is current; rotation is the magnetic field.
• Biot-Savart Law: Magnitude of field $dB$ due to element $dl$:     * $dB = \frac{\mu_0}{4\pi} \frac{I \, dl \sin(\theta)}{r^2}$     * In vector form: $d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{\hat{r}}}{r^2}$
• Applications:     * Long Straight Wire: $B = \frac{\mu_0 I}{2 \pi a}$     * Circular Coil (at Center): $B = \frac{\mu_0 N I}{2 R}$     * Circular Coil (Axial Point): $B = \frac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}$

Ampère’s Circuital Law

• Statement: The line integral of the magnetic field over a closed loop is $\mu_0$ times the net current enclosed.
• Mathematical Form: $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enclosed}$
• Applications:     * Solenoid: A long coil in helix form. Inside the solenoid, the field is uniform: $B = \mu_0 n I$ ($n$ is turns per unit length).     * Toroid: An endless solenoid (ring shape). Field inside the turns: $B = \mu_0 n I$. Field in open spaces (interior/exterior) is zero.

Lorentz Force and Motion of Charges

• Lorentz Force Equation: $\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$
• Magnetic Force: $\mathbf{F} = q(\mathbf{v} \times \mathbf{B})$. Magnitude $F = qvB\sin(\theta)$.     * Work done by magnetic Lorentz force is always zero. It changes direction, not speed.
• Motion in Uniform $B$ Field:     * Perpendicular Entrance ($\theta = 90^{\circ}$): Circular motion. Radius $r = \frac{mv}{qB}$. Frequency $f = \frac{qB}{2 \pi m}$.     * Oblique Entrance: Helical path due to component parallel to field.
• Velocity Selector: Perpendicular electric ($E$) and magnetic ($B$) fields allow charges with speed $v = \frac{E}{B}$ to pass undeflected.
• Cyclotron: Device to accelerate charged particles to high energies.     * Principle: Magnetic field makes ion move in circles; high-frequency electric field accelerates it across the gap between "Dees".     * Resonance Condition: Oscillator frequency must match cyclotron frequency ($f = f_{osc}$).     * Limitations: Cannot accelerate electrons (mass increase) or neutral particles.

Force on Conductors and Current Loops

• Force on Current-Carrying Wire: $\mathbf{F} = I(\mathbf{l} \times \mathbf{\mathbf{B}})$. (Fleming’s Left Hand Rule: Thumb-Force, Forefinger-Field, Middle-Current).
• Force Between Parallel Wires:     * Calculate force per unit length: $\frac{F}{l} = \frac{\mu_0 I_1 I_2}{2 \pi r}$     * Parallel currents attract; anti-parallel currents repel.     * Definition of Ampere: The current which, in two infinitely long parallel conductors 1m apart in vacuum, produces a force of $2 \times 10^{-7} \text{ N/m}$.
• Torque on a Rectangular Loop: $\mathbf{\tau} = \mathbf{p_m} \times \mathbf{B}$ where $\mathbf{p_m} = I \mathbf{A}$.     * For $N$ turns: $\tau = NABI \sin(\theta)$.
• Moving Coil Galvanometer:     * Deflection current: $I = \frac{K}{NAB} \theta = G \theta$     * Current Sensitivity: $S_I = \frac{\theta}{I} = \frac{NAB}{K}$.     * Voltage Sensitivity: $S_V = \frac{\theta}{V} = \frac{NAB}{KR_g}$.
• Conversion of Galvanometer:     * To Ammeter: Connect a low shunt resistance ($S$) in parallel. $S = \frac{I_g R_g}{I - I_g}$.     * To Voltmeter: Connect a high resistance ($R_h$) in series. $R_h = \frac{V}{I_g} - R_g$.