Magnetic Effect of Current and Magnetism

In 1820, Hans Christian Oersted discovered that an electric current deflects a nearby magnetic compass needle. Jean-Baptiste Biot and Félix Savart subsequently quantified the magnetic field produced by a steady line current element I d𝐥. At a displacement vector 𝐫 from the element, the infinitesimal magnetic field d𝐁 is given by the Biot-Savart Law:

where μ₀ = 4π × 10⁻⁷ T·m·A⁻¹ (or H/m) is the magnetic permeability of free space. In scalar magnitude:

where θ is the angle between the current element direction d𝐥 and the position vector 𝐫. The direction of d𝐁 is strictly perpendicular to the plane containing both d𝐥 and 𝐫, determined by the Right-Hand Screw Rule.

Consider a circular loop of radius R carrying steady current I in the y-z plane. For a point P on the loop axis (x-axis) at distance x from the center:

• The distance from any element d𝐥 on the perimeter to point P is r = √(R² + x²).
• Since d𝐥 is tangential to the circumference and 𝐫 lies in the plane connecting the element to the axis, d𝐥 ⊥ 𝐫, meaning sin 90° = 1.
• The magnetic field component perpendicular to the axis cancels out by diametrical symmetry: ∮ dB_⊥ = 0.
• The axial component is dB_x = dB cos α = dB · (R / r) = (μ₀ I dl / 4πr²) · (R / r) = (μ₀ I R dl) / [4π (R² + x²)^(3/2)].

Integrating around the entire circumference ∮ dl = 2πR for an N-turn coil:

Special Cases:

1. At the Center of the Loop (x = 0):
   B_center = (μ₀ N I) / (2R)
2. Far from the Loop (x ≫ R):
   B_far = (μ₀ N I R²) / (2x³) = (μ₀ / 4π) · [2 (N I · πR²) / x³] = (μ₀ / 4π) · (2M / x³)
   where M = N I A is the magnetic dipole moment. This proves that a planar current loop behaves identically to a magnetic dipole on its axial line!

Ampère's Circuital Law states that the line integral of the magnetic field 𝐁 around any closed arbitrary loop (called an Amperian loop) in free space equals μ₀ times the net steady current enclosed by the loop:

Sign Convention: Curl the fingers of the right hand in the direction of traversal along the closed Amperian loop; currents passing through the enclosed surface in the direction of the extended thumb are considered positive, while opposing currents are negative.

Consider an infinitely long, thin straight wire carrying steady current I. By cylindrical symmetry, magnetic field lines form concentric circles centered on the wire. Choose a circular Amperian loop of radius r concentric with the wire:

• At every point on the path, 𝐁 is tangential to the circle, so 𝐁 ∥ d𝐥 (cos 0° = 1).
• By symmetry, the magnitude of B is constant at all points equidistant from the wire.

A solenoid consists of a long helical winding of insulated wire. Let n = N / L represent the number of turns per unit length carrying current I. In an ideal long solenoid (L ≫ R):

1. The field outside the solenoid is practically zero: B_ext ≈ 0.
2. The field inside is uniform, intense, and strictly parallel to the solenoid axis.

Choose a rectangular Amperian loop abcd of length h, where side ab lies inside along the axis, sides bc and da extend radially outwards, and side cd lies outside:

The total current enclosed by the loop is I_enc = (n · h) · I. Applying Ampère's Law:

Important Boundary Note: At the extreme open ends of a semi-infinite solenoid, half the magnetic field lines escape into space, yielding exactly half the central value: B_end = (1/2) μ₀ n I.

A toroid is an endless solenoid bent into the shape of a doughnut (doughnut core of inner radius r₁ and outer radius r₂). With mean radius r and total turns N:

• Inside the Toroidal Core: Choosing a circular Amperian loop of radius r inside the core: ∮ 𝐁·d𝐥 = B (2πr) = μ₀ (N I) ⟹ B = (μ₀ N I) / (2πr) = μ₀ n I, where n = N / (2πr).
• In the Open Space Inside the Toroid (r < r₁): I_enc = 0 ⟹ B = 0.
• In the Space Outside the Toroid (r > r₂): Each turn enters and leaves the Amperian surface equally, so I_enc = N I - N I = 0 ⟹ B = 0.

When a particle carrying electric charge q moves with velocity 𝐯 through a region containing both an electric field 𝐄 and a magnetic field 𝐁, the net vector force experienced is the Lorentz Force:

Fundamental Properties of the Magnetic Force (𝐅_B = q(𝐯 × 𝐁)):

• Magnitude: F_B = |q| v B sin θ. The force vanishes if the particle is at rest (v = 0), uncharged (q = 0), or moves parallel/antiparallel to 𝐁 (θ = 0° or 180°).
• Direction: Determined by the right-hand cross product rule for positive charges, and inverted for negative charges (e.g. electrons).
• Zero Work Done: Since 𝐅_B ⊥ 𝐯 at every instant, the rate of work done is P = 𝐅_B · 𝐯 = q (𝐯 × 𝐁) · 𝐯 = 0. Therefore, a static magnetic field cannot alter the speed or kinetic energy of a charged particle—it acts purely as a deflecting agency!

1. Perpendicular Entry (θ = 90°): When 𝐯 ⊥ 𝐁, the constant normal force provides centripetal acceleration: q v B = m v² / r.
   Radius: r = (m v) / (q B) = p / (q B) = √(2 m K) / (q B)
   Time Period: T = (2πr) / v = (2πm) / (q B)
   Cyclotron Frequency: ν_c = 1 / T = (q B) / (2πm)
   Vital Exam Insight: Neither the orbital period T nor the cyclotron frequency ν_c depends on the velocity v or kinetic energy of the particle!
2. Oblique Entry (0° < θ < 90°): The velocity decomposes into v_∥ = v cos θ (parallel to 𝐁) and v_⊥ = v sin θ (perpendicular to 𝐁). The parallel component remains unaffected while the normal component causes circular motion, tracing a helix of radius r = (m v sin θ) / (q B) and linear pitch:
   Pitch = v_∥ · T = (v cos θ) · [(2πm) / (q B)] = (2πm v cos θ) / (q B)

When perpendicular electric and magnetic fields are arranged such that 𝐅_E = -𝐅_B, particles with a specific velocity pass through undeflected:

Invented by Ernest Lawrence and M. Stanley Livingston (1934), the Cyclotron accelerates positive ions (protons, deuterons, α-particles) to millions of electron-volts (MeV):

• Principle: Relies on the fact that the period of revolution of an ion inside a magnetic field is independent of its speed. An alternating radiofrequency (RF) electric field reverses polarity exactly when the ion completes a semicircle inside hollow D-shaped metal chambers ("Dees"), repeatedly accelerating the ion across the gap.
• Resonance Condition: The RF oscillator frequency f_osc must equal the cyclotron frequency: f_osc = ν_c = (q B) / (2πm).
• Maximum Kinetic Energy: If R is the radius of the Dees:
  v_max = (q B R) / m  ⟹  K_max = (1/2) m v_max² = (q² B² R²) / (2m)
• Limitations:
  1. Relativistic Mass Increase: At high velocities approaching speed of light c, relativistic mass increases: m = m₀ / √(1 - v²/c²). This causes the revolution period T to increase, throwing ions out of resonance with the fixed-frequency RF field.
  2. Cannot Accelerate Electrons: Electrons have very small rest mass, reaching relativistic speeds after modest acceleration, rapidly losing resonance.
  3. Cannot Accelerate Neutrons: Neutrons carry zero charge (q = 0) and cannot be accelerated by electric or magnetic fields.

A current I in a conductor of cross-sectional area A consists of mobile charge carriers (density n, charge q) drifting with average velocity 𝐯_d. The magnetic force on a segment d𝐥 containing n A dl carriers is:

For a straight conductor of uniform length 𝐋 in a uniform magnetic field 𝐁:

The direction is determined by Fleming's Left-Hand Rule: stretch the thumb, forefinger, and middle finger mutually perpendicular. If the Forefinger points along Field 𝐁, and the Middle finger represents Current I, then the Thumb gives the direction of Force (Motion) 𝐅.

Consider two infinitely long, thin parallel wires separated by distance d in vacuum, carrying steady currents I₁ and I₂:

• Wire 1 creates a magnetic field at the position of Wire 2: B₁ = (μ₀ I₁) / (2πd), directed into the page by the right-hand rule.
• Wire 2 experiences a force on length L due to B₁: F₂₁ = I₂ L B₁ sin 90° = (μ₀ I₁ I₂ L) / (2πd).
• By Newton's third law, Wire 1 experiences an equal and opposite force: 𝐅₁₂ = -𝐅₂₁.

Nature of the Force:

• Parallel Currents (Same Direction): Attract each other.
• Antiparallel Currents (Opposite Directions): Repel each other.

Consider a rectangular loop of dimensions a × b carrying current I in a uniform magnetic field 𝐁:

• Forces on opposite sides of length a are collinear, equal, and opposite, cancelling each other with zero net force and zero net torque.
• Forces on sides of length b have magnitude F = I b B. While they sum to zero net translational force (∑ 𝐅 = 0), their lines of action are separated by lever arm a sin θ (where θ is the angle between the normal vector 𝐧̂ of the loop and field 𝐁).

Defining the Magnetic Dipole Moment as 𝐌 = N I 𝐀 (where N is turn count, A is area, and 𝐀 points along right-hand normal):

Potential Energy of Dipole in Field:

• Stable Equilibrium (θ = 0°): U_min = -MB, τ = 0.
• Unstable Equilibrium (θ = 180°): U_max = +MB, τ = 0.

In Niels Bohr's atomic model, an electron (charge -e, mass m_e) revolves in a circular orbit of radius r with speed v:

• Equivalent electric current: I = e / T = e / (2πr / v) = (e v) / (2πr).
• Orbital magnetic moment: μ_l = I A = [(e v) / (2πr)] · (πr²) = (e v r) / 2.
• Relating to orbital angular momentum L = m_e v r:
  𝛍_l = - (e / 2m_e) 𝐋  ⟹  Gyromagnetic Ratio = |μ_l| / L = e / (2m_e) = 8.79 × 10¹⁰ C/kg
• Applying Bohr's quantization condition L = n ℏ = n (h / 2π) for the ground state (n = 1):
  Bohr Magneton: μ_B = (e ℏ) / (2m_e) = (e h) / (4π m_e) = 9.274 × 10⁻²⁴ A·m² (or J/T)

The Moving Coil Galvanometer is an electromagnetic instrument designed to detect and measure minuscule electric currents.

Essential Components:

1. A rectangular coil of fine insulated copper wire wound on an aluminum frame.
2. Cylindrical concave pole pieces of a permanent magnet that produce a radial magnetic field.
3. A central soft iron cylinder core that intensifies the field and ensures field lines remain strictly parallel to the coil plane at all rotational angles (θ = 90° always, so sin θ = 1).
4. A phosphor-bronze suspension strip providing restoring torque with small torsional constant k.

Working Principle: Deflecting torque τ_def = N I A B is counterbalanced by restoring torsional torque τ_res = k θ:

Gauss's Law for Magnetism states that the net magnetic flux Φ_B passing through any closed Gaussian surface is identically zero:

Physical Significance: Magnetic field lines are continuous closed loops with no beginning or end. Unlike electrostatics where isolated positive and negative charges exist, isolated magnetic monopoles do not exist in nature. Every magnet is an inseparable dipole.

The geomagnetic field approximates an imaginary magnetic dipole located at Earth's center, inclined at approximately 11.3° to the geographic rotational axis, with the magnetic South pole located near the geographic North pole. The field is fully specified at any location by three Magnetic Elements:

1. Magnetic Declination (θ or D): The angle between the geographic meridian and the magnetic meridian at a given point on Earth's surface.
2. Magnetic Inclination or Dip (δ or I): The angle between the direction of the total geomagnetic field 𝐁 and the horizontal plane in the magnetic meridian.
   • At the Magnetic Equator: δ = 0° (field is purely horizontal).
   • At the Magnetic Poles: δ = 90° (field is purely vertical).
3. Horizontal Component of Earth's Field (B_H):
   B_H = B cos δ
   B_V = B sin δ
   tan δ = B_V / B_H  ⟹  B = √(B_H² + B_V²)

When a material is subjected to an external magnetizing field 𝐇 (in A/m):

• Intensity of Magnetization (𝐌): Magnetic dipole moment induced per unit volume: 𝐌 = 𝐦_net / V (A/m).
• Magnetic Susceptibility (χ_m): Dimensionless ratio measuring material response: χ_m = M / H.
• Total Magnetic Field (Induction 𝐁): 𝐁 = μ₀ (𝐇 + 𝐌) = μ₀ 𝐇 (1 + χ_m) = μ 𝐇.
• Relative Magnetic Permeability (μ_r): μ_r = μ / μ₀ = 1 + χ_m.

Plotting magnetic flux density B versus magnetizing field H through a full cycle of AC magnetization reveals a closed Hysteresis Loop:

• Retentivity (Remanence): The residual magnetic induction B_r remaining in the specimen when magnetizing field H is reduced to zero.
• Coercivity: The reverse magnetizing field H_c required to completely demagnetize the specimen (reduce B to zero).
• Hysteresis Loss: The area enclosed by the B-H loop equals the thermal energy dissipated per unit volume per magnetization cycle: W = ∮ B dH.