Moving Charges and Magnetism - Comprehensive Study Notes

Historical Context and Oersted’s Discovery

• Long-standing Knowledge: Both electricity and magnetism have been recognized for over 2000 years. However, it was not until 1820 that they were realized to be fundamentally connected.
• Hans Christian Oersted (1820): During a lecture demonstration in the summer of 1820, the Danish physicist Oersted observed that a current in a straight wire caused a deflection in a nearby magnetic compass needle.
  • Needle Alignment: He found that the needle aligns tangentially to an imaginary circle centered on the wire, with the plane of the circle perpendicular to the wire.
  • Current and Proximity: The deflection becomes noticeable when the current is large or when the needle is sufficiently close to the wire, allowing the observer to neglect the Earth’s magnetic field.
  • Reversal and Magnitude: Reversing the current direction reverses the orientation of the needle. The deflection increases with higher current or reduced distance from the wire.
  • Iron Filings: When sprinkled around the wire, iron filings arrange themselves in concentric circles centered on the wire.
• Conclusion: Oersted concluded that moving charges or currents produce a magnetic field in the surrounding space.
• Scientific Advancement:
  • 1864: James Maxwell unified the laws of electricity and magnetism, realizing light consists of electromagnetic waves.
  • Late 19th Century: Radio waves were discovered by Hertz and produced by J.C. Bose and G. Marconi.
  • 20th Century: Progress accelerated due to understanding electromagnetism and inventions for the production, amplification, transmission, and detection of electromagnetic waves.

The Lorentz Force and Sources of Fields

• Electric Field (E) Recap:
  • A source charge $Q$ produces an electric field $E$:
  • E = \frac{Q \times \text{̒{r}}}{4̒̒̒̒_0 \times r^2}
  • A charge $q$ interacting with this field experiences a force: $F = q \times E$.
  • The field is a physical entity that conveys energy and momentum, propagating at a finite speed.
• Magnetic Field (B): Like the electric field, current or moving charges produce a magnetic field denoted by $B(r)$.
  • It is a vector field defined at each point in space and time.
  • Principle of Superposition: The magnetic field of several sources is the vector sum of the fields from each individual source.
• Lorentz Force Equation: The total force on a charge $q$ moving with velocity $v$ in the presence of both electric and magnetic fields is given by:
  • $F = q [ E(r) + v \times B(r)] = F_{electric} + F_{magnetic}$
• Features of Magnetic Force ($F_{magnetic} = q [ v \times B ]$):
  • It depends on the charge $q$, velocity $v$, and magnetic field $B$. The force on a negative charge is opposite to that on a positive charge.
  • Vector Product: The force vanishes if the velocity and magnetic field are parallel or anti-parallel (angle is 0° or 180°).
  • Direction: The force acts perpendicularly to both the velocity and the magnetic field. Its direction is determined by the Right-Hand Rule or the Screw Rule.
  • Static Charges: The magnetic force is zero if the charge is not moving ($|v| = 0$). Only moving charges experience magnetic force.

Magnetic Field Units and Constants

• Magnitude Definition: The magnitude of magnetic field $B$ is 1 SI unit when the force on 1 Coulomb ($1‐C$) moving perpendicular to $B$ at $1‐m/s$ is 1 Newton ($1‐N$).
• Dimensional Formula: $[B] = [F / (q \times v)]$
• Units:
  • SI Unit: Tesla ($T$), named after Nikola Tesla (1856–1943). Also expressed as Newton•second / (coulomb•meter).
  • CGS Unit: Gauss ($G$). $1‐G = 10^{-4}‐T$.
  • Earth’s Magnetic Field: Approximately $3.6 \times 10^{-5}‐T$ or $0.36‐G$.
• Permeability of Free Space (̒_0):
  • \frac{̒_0}{4̒} = 10^{-7}‐T \times m/A.
• Relation to Light Speed: The speed of light in vacuum $c$ is related to permittivity (̒_0) and permeability (̒_0) by:
  • ̒_0 \times ̒_0 = \frac{1}{c^2}

Magnetic Force on a Current-Carrying Conductor

• Derivation: Consider a rod of cross-sectional area $A$, length $l$, and number density of mobile carriers $n$.
  • Total charge carriers: $(n \times l \times A)$.
  • Force on carriers in field $B$: $F = (n \times l \times A) \times q \times v_d \times B$, where $v_d$ is the drift velocity.
  • Since current density $j = n \times q \times v_d$ and current $I = |j| \times A$, the force simplifies to:
  • $F = I \times l \times B$
  • Here, $l$ is a vector with magnitude equal to the length of the rod and direction matching the current flow.
• Arbitrary Shapes: If a wire has an irregular shape, the total force is the integral of infantisimal elements: $F = ∑ I \times dl_j \times B \rightarrow ∫ I \times dl \times B$.
• Example 4.1: A straight wire ($200‐g$, $1.5‐m$) carrying $2‐A$ is suspended in mid-air by a horizontal magnetic field $B$.
  • Force balance: $m \times g = I \times l \times B$
  • $B = \frac{m \times g}{I \times l} = \frac{0.2 \times 9.8}{2 \times 1.5} = 0.65‐T$.

Motion of Charged Particles in Magnetic Fields

• Work Done: Since the magnetic force is always perpendicular to velocity ($F ∕ v$), the work done by the magnetic field on a charge is zero ($W = 0$). The magnitude of velocity (kinetic energy) remains constant, but the direction changes.
• Circular Motion ($v ∕ B$):
  • The magnetic force acts as a centripetal force: $\frac{m \times v^2}{r} = q \times v \times B$.
  • Radius: $r = \frac{m \times v}{q \times B}$.
  • Angular Frequency (̒): ̒ = \frac{v}{r} = \frac{q \times B}{m}.
  • Frequency (̒): ̒ = \frac{q \times B}{2̒ \times m}.
  • Time Period ($T$): T = \frac{2̒ \times m}{q \times B}.
  • Crucially, frequency and period are independent of the particle’s speed and radius.
• Helical Motion: If the velocity has a component parallel to the magnetic field (v_{||} = v \times \text{cos}(̒)), the particle moves along the field lines while rotating, creating a helix.
  • Pitch ($p$): The distance moved along the field in one rotation:
  • p = v_{||} \times T = \frac{2̒ \times m \times v_{||}}{q \times B}.

Biot-Savart Law

• Definition: This law relates the magnetic field $dB$ at a point $P$ to an infinitesimal current element $I \times dl$ at distance $r$.
  • dB = \frac{̒_0}{4̒} \times \frac{I \times dl \times r}{r^3}
  • Vector Form Magnitude: dB = \frac{̒_0}{4̒} \times \frac{I \times dl \times \text{sin}(̒)}{r^2}.
• Comparison to Coulomb’s Law:
  1. Similarities: Both are long-range ($1/r^2$ dependence) and obey the principle of superposition.
  2. Source: Electric field is produced by a scalar (charge); magnetic field is produced by a vector (current element $I \times dl$).
  3. Direction: Electric field is along the displacement vector; magnetic field is perpendicular to the plane of $dl$ and $r$.
  4. Angle Dependence: Biot-Savart law depends on the angle ̒ between $dl$ and $r$; the field is zero along the line of the current element.

Magnetic Field of a Circular Current Loop

• On-Axis Calculation: For a loop of radius $R$ carrying current $I$, at distance $x$ from the center along the axis:
  • B = \frac{̒_0 \times I \times R^2}{2(x^2 + R^2)^{3/2}}
• At the Center ($x = 0$):
  • B = \frac{̒_0 \times I}{2 \times R}
• Field Lines: The field lines form closed loops. The direction is given by the Right-Hand Thumb Rule: Curl fingers in the direction of the current, and the thumb indicates the direction of the magnetic field.
• Example 4.6: A 100-turn coil of radius $10‐cm$ carrying $1‐A$ current:
  • B = \frac{100 \times ̒_0 \times I}{2 \times R} = 6.28 \times 10^{-4}‐T.

Ampere’s Circuital Law

• Statement: The line integral of the magnetic field $B$ around a closed loop is equal to ̒_0 times the total current $I$ passing through the surface enclosed by the loop:
  • ∮ B ⋅ dl = ̒_0 \times I
• Right-Hand Rule for Signs: Curl fingers in the direction of the integral traversal; the thumb indicates the positive direction for current.
• Solenoid Field: A solenoid is a long wire wound into a helix. Inside a long solenoid, the field is uniform and parallel to the axis; reaching outside, the field is nearly zero.
  • B = ̒_0 \times n \times I
  • Here, $n$ is the number of turns per unit length.
• Straight Infinite Wire: Applying Ampere's Law to a circle of radius $r$ surrounding a wire:
  • B \times (2̒ \times r) = ̒_0 \times I \rightarrow B = \frac{̒_0 \times I}{2̒ \times r}.

Force Between Two Parallel Currents

• Parallel Currents: Two long parallel conductors separated by distance $d$ carrying currents $I_a$ and $I_b$.
  • Field due to 'a' at 'b': B_a = \frac{̒_0 \times I_a}{2̒ \times d}.
  • Force on length $L$ of 'b': F_{ba} = I_b \times L \times B_a = \frac{̒_0 \times I_a \times I_b \times L}{2̒ \times d}.
• Directional Strength:
  • Parallel currents attract.
  • Anti-parallel currents repel.
• Definition of the Ampere (SI Unit): One Ampere is the steady current which, when maintained in two infinitely long parallel conductors 1 meter apart in vacuum, produces a force of $2 \times 10^{-7}‐N/m$ on each conductor.

Torque on Current Loops and Galvanometers

• Torque Calculation: A rectangular loop of area $A$ carrying current $I$ in a uniform field $B$ experiences a torque:
  • ̒ = m \times B
  • Where $m$ is the magnetic moment: $m = N \times I \times A$.
  • Magnitude: ̒ = N \times I \times A \times B \times \text{sin}(̒).
• Moving Coil Galvanometer (MCG):
  • Principle: A current-carrying coil in a magnetic field experiences a torque that is balanced by a spring's restoring torque.
  • Equilibrium: k \times ̒ = N \times I \times A \times B, where $k$ is the torsional constant of the spring.
  • Deflection: ̒ = (\frac{N \times A \times B}{k}) \times I.
  • Radial Field: Achieved via a cylindrical soft iron core, ensuring \text{sin}(̒) = 1.
• Instrument Conversions:
  • Ammeter: Connect a small shunt resistance ($r_s$) in parallel with the galvanometer.
  • Voltmeter: Connect a large resistance ($R$) in series with the galvanometer.
• Sensitivity:
  • Current Sensitivity: \frac{̒}{I} = \frac{N \times A \times B}{k}.
  • Voltage Sensitivity: \frac{̒}{V} = \frac{N \times A \times B}{k \times R_{galvanometer}}.
  • Note: Doubling the number of turns doubles current sensitivity but might not change voltage sensitivity as the resistance also increases.