NCERT Physics Class 12 Chapter 4: Moving Charges and Magnetism Study Notes

Historical Context and the Discovery of Electromagnetism

• Long-Term Knowledge: Both electricity and magnetism have been observed for over 2,000 years, yet their intrinsic relationship was only established in 1820.

• Oersted's Discovery: Danish physicist Hans Christian Oersted (1777–1851) was the first to provide empirical evidence linking the two during a lecture demonstration in the summer of 1820.

  • Observations: He witnessed a current in a straight wire causing a deflection in a nearby magnetic compass needle.

  • Field Orientation: Oersted found that the needle aligns tangentially to an imaginary circle centered on the wire, with the plane of the circle being perpendicular to the wire.

  • Determinants of Deflection: The deflection increases as the current is increased or if the needle is moved closer to the wire. Reversing the current direction reverses the orientation of the needle.

  • Visualization: Iron filings sprinkled around such a wire arrange themselves in concentric circles representing the magnetic field lines.

• Unification and Progress:

  • James Maxwell (1864): Integrated the laws of electricity and magnetism into a unified framework and realized that light consists of electromagnetic waves.

  • Hertz, Bose, and Marconi: By the end of the 19th century, radio waves were discovered (Hertz) and later produced for communication (J.C. Bose and G. Marconi).

Fundamental Concepts of Magnetic Fields and Force

• The Field Concept:

  • Fields are not just mathematical artifacts; they have a physical role, conveying energy and momentum.

  • Fields do not establish themselves instantaneously; they take a finite time to propagate.

  • Static charges produce an electric field ($\mathbf{E}$), while moving charges or currents produce both an electric field ($\mathbf{E}$) and a magnetic field ($\mathbf{B}$).

• Conventions for Field Direction:

  • Dot ($\cdot$): Represents a current or field emerging out of the plane of the paper (like the tip of an arrow pointed at the viewer).

  • Cross ($\otimes$): Represents a current or field going into the plane of the paper (like the feathered tail of an arrow moving away).

• Superposition Principle: Like electric fields, magnetic fields at any point in space add vectorially if there are multiple sources.

• Lorentz Force: The total force ($\mathbf{F}$) on a point charge $q$ moving with velocity $\mathbf{v}$ in the presence of an electric field $\mathbf{E}$ and a magnetic field $\mathbf{B}$ is:

  • $\mathbf{F} = q [\mathbf{E}(\mathbf{r}) + \mathbf{v} \times \mathbf{B}(\mathbf{r})] \equiv \mathbf{F}_{electric} + \mathbf{F}_{magnetic}$

  • This formulation was provided by Hendrik Antoon Lorentz (1853–1928), based on experiments by Ampere and others.

Characteristics of the Magnetic Force

• Dependencies: The force depends on the charge ($q$), velocity ($\mathbf{v}$), and the magnetic field ($\mathbf{B}$). The force on a negative charge is opposite to that on a positive charge.

• Vector Product Properties: The force involves a cross product ($\mathbf{v} \times \mathbf{B}$).

  • Zero Force Situations: The magnetic force vanishes if the velocity is parallel or anti-parallel to the magnetic field, or if the charge is stationary ($|v| = 0$).

  • Direction: The force acts in a "sideways" direction, perpendicular to both $\mathbf{v}$ and $\mathbf{B}$. Its direction is determined by the right-hand rule or screw rule.

• Magnitude: The magnitude of the magnetic force is given by:

  • $F = q v B \sin(\theta)$

  • where $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{B}$.

• Units of Magnetic Field:

  • SI Unit: The tesla ($T$), named after Nikola Tesla (1856–1943). One Tesla is defined as the magnitude of $B$ when a charge of $1\,C$ moving perpendicular to $B$ at $1\,m/s$ feels a force of $1\,N$.

  • Dimensions: $[B] = [F / qv] = [M T^{-2} A^{-1}]$.

  • Non-SI Unit: The gauss ($G$), where $1\,G = 10^{-4}\,T$.

  • Example Magnitudes:

    • Surface of a neutron star: $10^{8}\,T$

    • Earth's magnetic field: $3.6 \times 10^{-5}\,T$

    • Human nerve fiber: $10^{-10}\,T$

Magnetic Force on a Current-Carrying Conductor

• Derivation: For a straight rod of cross-sectional area $A$, length $l$, and mobile charge carrier density $n$, carrying a steady current $I$:

  • The total number of carriers is $nAl$.

  • If each carrier has a drift velocity $\mathbf{v}_d$, the force is $\mathbf{F} = (nAl) q \mathbf{v}_d \times \mathbf{B}$.

  • Since the current density $\mathbf{j} = n q \mathbf{v}_d$ and current $I = |j|A$, we have:

  • $\mathbf{F} = I \mathbf{l} \times \mathbf{B}$

• Vector Definition: In the equation $\mathbf{F} = I \mathbf{l} \times \mathbf{B}$, $\mathbf{l}$ is a vector with magnitude equal to the length of the rod and direction identical to the current flow (though current itself is a scalar).

• Arbitrary Shapes: For wires of non-linear geometry, the force is the sum/integral of forces on infinitesimal segments $d\mathbf{l}$:

  • $\mathbf{F} = \int I d\mathbf{l} \times \mathbf{B}$

Permittivity and Permeability

• Definitions:

  • Electric Permittivity ($\epsilon$): Describes how an electric field affects and is affected by a medium (related to material polarization).

  • Magnetic Permeability ($\mu$): The ability of a substance to acquire magnetization; a measure of how much a magnetic field can penetrate matter.

• Medium Dependence: Unlike the gravitational constant $G$, constants $\epsilon$ and $\mu$ are not universal and depend on the medium.

• Relationship to Light: The product of permittivity and permeability is related to the speed of light ($v$) in that medium:

  • $\epsilon \mu = \frac{1}{v^2}$

• Vacuum Values: In vacuum, the exact value for permeability $\mu_0/4\pi = 10^{-7}\,T\,m/A$.

Motion of a Charged Particle in a Magnetic Field

• Work and Energy: The magnetic force is always perpendicular to the velocity ($\mathbf{F} \cdot \mathbf{v} = 0$). Therefore, the magnetic field does no work on the charge, and the magnitude of velocity (kinetic energy) remains constant, though direction changes.

• Circular Motion ($\mathbf{v} \perp \mathbf{B}$): When the velocity is perpendicular to a uniform magnetic field, the Lorentz force acts as a centripetal force:

  • $\frac{m v^2}{r} = q v B \implies r = \frac{m v}{q B}$

  • Radius ($r$): Proportional to momentum ($mv$).

  • Angular Frequency ($\omega$): $\omega = \frac{v}{r} = \frac{q B}{m}$.

  • Frequency ($\nu$): $\nu = \frac{q B}{2 \pi m}$. Crucially, this frequency is independent of the particle's speed or energy.

• Helical Motion: If the velocity has a component parallel to the field ($v_{\parallel}$), that component remains constant while the perpendicular component ($v_n$) behaves circularly.

  • The resulting path is a helix.

  • Pitch ($p$): The distance moved along the field lines in one rotation: $p = v_{\parallel} T = \frac{2 \pi m v_{\parallel}}{q B}$.

• Natural Phenomenon: Aurora Borealis: Solar flares eject electrons and protons that get trapped by the Earth's magnetic field and follow helical paths toward the poles. Collisions with atmospheric atoms cause light emissions: Green (Oxygen) and Pink (Nitrogen).

Motion in Combined Electric and Magnetic Fields

• Velocity Selector: In a region where $\mathbf{E}$ and $\mathbf{B}$ are perpendicular to each other and to the particle's velocity (crossed fields), the total force is zero if the magnitudes of the electric and magnetic forces are equal:

  • $q E = q v B \implies v = \frac{E}{B}$

  • Only particles moving at this specific speed pass through undeflected.

• Cyclotron: A machine used to accelerate ions to high energies.

  • Components: Two semi-circular metal containers called "dees" ($D_1, D_2$) in an evacuated chamber, subject to a uniform magnetic field.

  • Mechanism: A high-frequency alternating voltage is applied to the dees. Ions are accelerated across the gap. Inside the dees, the magnetic field makes them follow a circular path without an electric field.

  • Resonance Condition: The oscillator frequency $\nu_a$ must match the cyclotron frequency $\nu_c = \frac{q B}{2 \pi m}$.

  • Maximum Energy: Kinetic energy at exit radius $R$ is:

    • $K = \frac{q^2 B^2 R^2}{2 m}$

The Biot-Savart Law

• Equation: Describes the magnetic field $d\mathbf{B}$ produced by an infinitesimal current element $I d\mathbf{l}$:

  • $d\mathbf{B} = \frac{\mu_0}{4 \pi} \frac{I d\mathbf{l} \times \mathbf{r}}{r^3}$

  • Magnitude: $dB = \frac{\mu_0}{4 \pi} \frac{I dl \sin(\theta)}{r^2}$

• Comparison to Coulomb's Law:

  • Similarities: Both are long-range ($1/r^2$ dependence) and obey the principle of superposition.

  • Differences: Electrostatic source is a scalar ($q$), magnetic source is a vector ($I d\mathbf{l}$). Magnetic fields are perpendicular to the displacement vector, and there is an angular dependence ($\sin(\theta)$).

Magnetic Field of a Circular Current Loop

• Field on Axis: For a loop of radius $R$ with current $I$, the field at distance $x$ along the axis from the center is:

  • $B = \frac{\mu_0 I R^2}{2 (x^2 + R^2)^{3/2}}$

  • Direction: Along the axis, determined by the right-hand thumb rule (curling fingers in the direction of current points the thumb toward the field direction).

• Field at Center ($x = 0$):

  • $B = \frac{\mu_0 I}{2 R}$

• Loop as a Dipole: At large distances ($x >> R$), the field behaves like an electric dipole:

  • $B \approx \frac{\mu_0 2 m}{4 \pi x^3}$, where $m = I A$ is the magnetic moment.

Ampere's Circuital Law

• Definition: The line integral of the magnetic field $\mathbf{B}$ around a closed loop is equal to $\mu_0$ times the total currentIpassing through the surface enclosed by the loop:

  • $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I$

• Application to Symmetry:

  • Straight Wire: At distance $r$ from an infinite wire, $B(2 \pi r) = \mu_0 I \implies B = \frac{\mu_0 I}{2 \pi r}$.

  • Inside a Wire (Radius $a$, uniformly distributed current):

    • For $r > a$: $B = \frac{\mu_0 I}{2 \pi r}$.

    • For $r < a$: $B = \frac{\mu_0 I r}{2 \pi a^2}$ (linear increase with distance).

Solenoids and Toroids

• Long Solenoid: A long wire wound into a close helix.

  • Field Inside: Strong, uniform, and along the axis: $B = \mu_0 n I$, where $n$ is turns per unit length ($n = N/L$).

  • Field Outside: Effectively zero for an ideal/long solenoid.

• Toroid: A solenoid bent into the shape of a hollow circular ring.

  • Field Inside the Core: $B = \frac{\mu_0 N I}{2 \pi r} = \mu_0 n I$.

  • Open Spaces: The field is zero in the open space inside the ring and outside the toroid.

Force and Torque between Current Systems

• Parallel Currents: Two long conductors separated by distance $d$ carry currents $I_a$ and $I_b$.

  • Force per Unit Length ($f$): $f = \frac{\mu_0 I_a I_b}{2 \pi d}$.

  • Rule: Parallel currents attract; anti-parallel (opposite) currents repel.

  • Definition of Ampere: One ampere is the current which, if maintained in two straight parallel conductors 1 meter apart in vacuum, produces a force of $2 \times 10^{-7}\,N/m$ on each conductor.

• Torque on a Rectangular Loop: A loop of area $A$ with current $I$ in a uniform field $\mathbf{B}$ experiences a torque:

  • $\tau = \mathbf{m} \times \mathbf{B} = m B \sin(\theta)$

  • The magnetic moment $\mathbf{m} = N I \mathbf{A}$.

  • Equilibrium: Stable when $\mathbf{m}$ and $\mathbf{B}$ are parallel; unstable when anti-parallel.

Atomic Magnetism and the Moving Coil Galvanometer

• Revolving Electron: An electron revolving around a nucleus acts as a current loop.

  • Magnetic Moment ($\mu_l$): $\mu_l = \frac{e}{2 m_e} l$, where $l$ is orbital angular momentum.

  • Gyromagnetic Ratio: $\frac{\mu_l}{l} = \frac{e}{2 m_e} \approx 8.8 \times 10^{10}\,C/kg$.

  • Bohr Magneton: The minimum dipole moment ($n=1$ in Bohr model): $\mu_B = \frac{e h}{4 \pi m_e} \approx 9.27 \times 10^{-24}\,A m^2$.

• Moving Coil Galvanometer (MCG): Measures small currents using torque on a coil.

  • Principle: Torque $NIAB$ is balanced by a spring counter-torque $k \phi$.

  • Current Sensitivity: $\phi / I = \frac{NAB}{k}$.

  • Modifications:

    • Ammeter: Created by adding a small shunt resistance ($r_s$) in parallel.

    • Voltmeter: Created by adding a large series resistance ($R$).

Summary of Physical Quantities

• Permeability of Free Space ($\mu_0$): $[M L T^{-2} A^{-2}]$, Unit: $T\,m\,A^{-1}$, Value: $4\pi \times 10^{-7}$.

• Magnetic Field ($B$): $[M T^{-2} A^{-1}]$, Unit: tesla ($T$).

• Magnetic Moment ($m$): $[L^2 A]$, Unit: $A\,m^2$ or $J/T$.

• Torsion Constant ($k$): $[M L^2 T^{-2}]$, Unit: $N\,m\,rad^{-1}$.

Questions & Discussion

• Work by Magnetic Fields: Why does the kinetic energy of a particle not change in a magnetic field?

  • Answer: Because the magnetic force ($q \mathbf{v} \times \mathbf{B}$) is always perpendicular to the direction of motion, meaning the dot product $\mathbf{F} \cdot d\mathbf{r}$ is zero; thus, no work is performed.

• Galvanometer Conversion Logic: How can current sensitivity be doubled without changing voltage sensitivity?

  • Answer: Increasing the number of turns ($N$) to $2N$ doubles current sensitivity, but it also usually doubles the resistance ($R$). Since voltage sensitivity depends on $N/R$, it remains unchanged.

• Roget's Spiral Experiment: A vertical spring's tip touches a mercury surface in a circuit. When current flows, parallel turns of the spring attract each other, pulling the tip out of the mercury and breaking the circuit. The cycles repeats as the spring relaxes, creating a "tick, tick" oscillation.