Physics 202 - Chapter 30: Sources of Magnetic Fields

Chapter 30: Sources of Magnetic Fields

Outline

• Biot-Savart Law
• Magnetic force between two conductors
• Ampere's Law
• Magnetic field of a solenoid
• Gauss's Law in Magnetism
• Magnetism in Matter

Summary

• Biot-Savart Law:
  • The magnetic field dB at a point P due to a length element ds carrying a steady current I is given by: dB = aivefrac{μ_0}{4π} aivefrac{Ids × r}{r^2}
    • μ_0 is the permeability of free space.
    • r is the distance from the element to the point P.
    • ŕ is a unit vector pointing from ds toward point P.
  • The total field at P is found by integrating this expression over the entire current distribution:
    B =
    aivefrac{μ_0 I}{4π} ∫
    aivefrac{ds × ŕ}{r^2}
• Magnetic Force Between Two Parallel Wires:
  • The magnetic force per unit length between two parallel wires separated by a distance a and carrying currents I1 and I2 is:
    F/ℓ =
    aivefrac{μ0 I1 I_2}{2πa}
  • The force is attractive if the currents are in the same direction and repulsive if they are in opposite directions.
• Ampère's Law:
  • The line integral of B ⋅ ds around any closed path equals μ0I, where I is the total steady current through any surface bounded by the closed path: ∮ B ⋅ ds = μ0I
• Magnetic Field of a Long, Straight Wire:
  • The magnitude of the magnetic field at a distance r from a long, straight wire carrying an electric current I is:
    B =
    aivefrac{μ_0I}{2πr}
  • The field lines are circles concentric with the wire.
• Magnetic Field Inside a Toroid:
  • The magnetic field inside a toroid is:
    B =
    aivefrac{μ_0NI}{2πr}, where N is the total number of turns.
• Magnetic Field Inside a Solenoid:
  • The magnetic field inside a solenoid is:
    B = μ0 aivefrac{N}{L} I = μ0nI, where N is the total number of turns, L is the length, and n is the number of turns per unit length.
• Gauss's Law for Magnetism:
  • The net magnetic flux through any closed surface is zero:
    ∮ B ⋅ dA = 0
• Magnetic Materials:
  • Diamagnetic Substances: Weak magnetic moment, opposite to the applied magnetic field.
  • Paramagnetic Substances: Weak magnetic moment, in the same direction as the applied magnetic field.
  • Ferromagnetic Substances: Interactions between atoms cause magnetic moments to align, creating a strong magnetization that remains after the external field is removed.

Biot-Savart Law Details

• Experiments showed:
  • The vector dB is perpendicular to both ds (direction of the current) and the unit vector ŕ (from ds toward P).
  • The magnitude of dB is inversely proportional to r^2, where r is the distance from ds to P.
  • The magnitude of dB is proportional to the current I and the magnitude of ds.
  • The magnitude of dB is proportional to sin θ, where θ is the angle between the vectors ds and ŕ.
• Mathematical Formulation: dB = aivefrac{μ_0}{4π} aivefrac{I ds × ŕ}{r^2}
  • ds is in the direction of the current.
  • μ0 is the permeability of free space: μ0 = 4π × 10^{-7} T ⋅ m/A
• The total field is obtained by integrating over the whole current distribution:
  B =
  aivefrac{μ_0 I}{4π} ∫
  aivefrac{ds × ŕ}{r^2}
• The direction of B is determined by the right-hand rule.

Magnetic Field vs. Electric Field

• You must integrate over the entire current distribution for magnetic fields.

Biot-Savart Law - Examples

• Straight Wire:
  • Consider a thin, straight wire carrying a constant current I along the x-axis.
  • For an infinite wire, the magnetic field is calculated with integration limits θ1 = π/2 and θ2 = -π/2.
• Current-Carrying Wire Segment:
  • Calculate the magnetic field at point O for a wire segment with two straight portions and a circular arc of radius a subtending an angle θ.
  • For segments A’A and C’C, ds × r = 0.
  • Along the arc, |ds × r| = ds, and dB is into the page.
  • The B field at the center of a circular loop is thus (θ → 2π).
• Circular Wire Loop:
  • Consider a circular wire loop of radius a in the yz-plane carrying a steady current I.
  • Calculate the magnetic field at an axial point P a distance x from the loop's center.
  • For all ds along the loop, |ds × r| = ds, and dB is perpendicular to the loop's axis.
  • The magnetic field at the center of the loop (x = 0) is:
    B =
    aivefrac{μ_0I}{2a}
  • For x >> a, the magnetic field is given by:
    B ≈
    aivefrac{μ_0}{4π}
    aivefrac{2μ}{x^3}, where μ = Iπa^2 is the dipole moment of the loop.
  • This is similar to the electric field due to an electric dipole.

Magnetic Force Between Two Parallel Conductors

• If the force per unit length between two long, parallel wires carrying identical currents separated by 1 m is 2 × 10^{-7} N/m, the current in each wire is defined as 1 A.
• The field B2 due to the current in wire 2 exerts a force on wire 1 of: F1 = I1ℓB2.
• The magnetic field of wire 2 is: B2 = aivefrac{μ0I_2}{2πa}.
• Thus, the force per unit length is: F/ℓ =
  aivefrac{μ0I1I_2}{2πa}.
• The force is attractive for parallel currents and repulsive for antiparallel currents.

Ampère’s Law

• The magnetic field lines are circles concentric with the wire.
• The field lines lie in planes perpendicular to the wire.
• The magnitude of the field is constant on any circle of radius a.
• The right-hand rule determines the field's direction.
• Evaluating the integral of B ⋅ ds over a circular path around the current:
  ∮ B ⋅ ds = B ∮ ds = B(2πr) = μ0I Thus, B = aivefrac{μ0I}{2πr}.
• Ampère’s law describes the creation of magnetic fields by all continuous current configurations.
• It is used for calculating the magnetic field of current configurations with a high degree of symmetry,
• Similar to Gauss’s law in calculating electric fields for highly symmetric charge distributions.

Ampère’s Law – Examples

• Magnetic Field Due to a Long Current-Carrying Wire:
  • A long, straight wire of radius R carries a steady current I uniformly distributed through the cross-section.
  • Calculate the magnetic field at a distance r from the wire's center for r >= R and r < R.
  • For r >= R:
    ∮ B ⋅ ds = μ0I B(2πr) = μ0I
    B =
    aivefrac{μ_0I}{2πr}
  • For r < R:
    ∮ B ⋅ ds = μ0I{enc}
    B(2πr) = μ0I aivefrac{πr^2}{πR^2} B = aivefrac{μ0Ir}{2πR^2}
• Magnetic Field Due to a Solenoid:
  • A solenoid is a long wire wound in the form of a helix.
  • A reasonably uniform magnetic field can be produced in the space surrounded by the wire turns.
  • The field lines in the interior are nearly parallel, uniformly distributed, and close together, indicating a strong and almost uniform field.
  • Applying Ampère’s law to a rectangle with side ℓ parallel to the interior field:
    ∮ B ⋅ ds = ∫1 B ⋅ ds + ∫2 B ⋅ ds + ∫3 B ⋅ ds + ∫4 B ⋅ ds = μ0NI Bℓ = μ0nℓI
    B = μ_0nI where n is the number of turns per unit length.
  • The field distribution is similar to that of a bar magnet.
  • An ideal solenoid has closely spaced turns and a length much greater than the radius of the turns.

Magnetic Flux

• The magnetic flux associated with a magnetic field is defined similarly to electric flux.
• Consider an area element dA on an arbitrarily shaped surface.
• The magnetic flux through the surface is: Φ_B = ∫ B ⋅ dA.
• If the magnetic field makes an angle theta with the area vector, then the flux is: Φ_B = B A cos θ.
• The unit of the magnetic flux is the Weber (Wb): T ⋅ m^2 = Wb.

Gauss's Law in Magnetism

• Magnetic fields do not begin or end at any point.
• Magnetic field lines are continuous and form closed loops.
• The number of lines entering a surface equals the number of lines leaving the surface.
• Therefore, the integral of the magnetic field over any closed surface is zero: ∮ B ⋅ dA = 0.
• No magnetic monopoles have been found (i.e., no separate North or South).

Magnetism in Matter

• Semi-classical theory: an electron as a point charge orbiting a positive nucleus with radius r and periodic time T =
  aivefrac{2πr}{v}.
• The movement of the electron amounts to a current loop.
• The current in this loop is: I =
  aivefrac{e}{T} =
  aivefrac{ev}{2πr}.
• The orbital angular momentum of the electron is L = m_e v r.
• The magnetic moment of the loop is:
  μ = IA =
  aivefrac{ev}{2πr} πr^2 =
  aivefrac{evr}{2} =
  aivefrac{e}{2me} (me vr) =
  aivefrac{e}{2m_e} L
• The magnetic moment of the atom is also quantized.
• The smallest value of μ is the Bohr magneton: μB = aivefrac{eħ}{2me} = 9.27 × 10^{-24} J/T.

Magnetism in Matter – Spin Magnetic Moment

• The electron has an intrinsic property called spin, which gives it another magnetic moment.
• The spin angular momentum is also quantized: S =
  aivefrac{\sqrt{3}}{2} ħ.
• The magnetic moment associated with this spin is: μ{spin} = aivefrac{eħ}{2me}.
• The total magnetic moment of an atom is the vector sum of the orbital and spin magnetic moments of all its electrons: μ{atom} = ∑{all electrons} (μ + μ_{spin}).

Magnetism in Matter – Magnetization

• Magnetization Vector: M =
  aivefrac{μ}{V}, where μ is the total magnetic moment and V is the volume.
• The magnetization produces its own magnetic field: BM = μ0M.
• If there is also an external magnetic field B0, then the total field is: B = B0 + BM = μ0(H + M), where H =
  aivefrac{B0}{μ0} is the magnetic field strength of the external field B_0.

Magnetism in Matter – Materials Response

• The magnetization of a material is proportional to the external field strength H: M = χH, where χ is the magnetic susceptibility of the material.
• The total magnetic flux density B inside the material:
  B = μ0(H + M) = μ0(H + χH) = μ_0(1 + χ)H
• The magnetic permeability of a material is defined as: μm = μ0(1 + χ)
• Thus, the total magnetic flux density inside a material due to an external field is: B = μ_m H

Magnetic Materials

• Diamagnetic Material: B < H
• Paramagnetic Material: B > H
• Ferromagnetic Material: B >> H

Ferromagnetic Materials

• Microscopic regions (domains) have aligned magnetic moments in different directions.
• When B_0 = 0 and M = 0 due to random orientation of the domains.
• When B0 > 0, domains parallel to B0 grow.
• After the field is removed, the domains remain oriented, so the material remains magnetized and forms a magnet.
• A magnet can be demagnetized by reversing an external field in cycles to restore random domains.

Paramagnetic Materials

• Paramagnetic substances have a weak magnetism resulting from the presence of atoms (or ions) that have permanent magnetic moments.
• These moments interact only weakly with one another and are randomly oriented in the absence of an external magnetic field.
• When a paramagnetic substance is placed in an external magnetic field, its atomic moments tend to line up with the field.
• This alignment competes with thermal motion, which tends to randomize the magnetic moment orientations.

Diamagnetic Materials

• When an external magnetic field is applied to a diamagnetic substance, a weak magnetic moment is induced in the direction opposite the applied field, causing diamagnetic substances to be weakly repelled by a magnet.
• Although diamagnetism is present in all matter, its effects are much smaller than those of paramagnetism or ferromagnetism and are evident only when those other effects do not exist.

Magnetic Materials – χ Values

• Paramagnetic materials: χ > 0 => μm > μ0
• Diamagnetic materials: χ < 0 => μm < μ0

Meissner Effect

• Certain types of superconductors also exhibit perfect diamagnetism in the superconducting state.
• This is called the Meissner effect.
• If a permanent magnet is brought near a superconductor, the two objects repel each other.

Ampère's Law - Quizzes

• Quick Quiz 30.3: Ranking magnitudes of ∮ B ⋅ ds for closed paths a through d.
• Quick Quiz 30.4: Ranking magnitudes of ∮ B ⋅ ds for closed paths a through d near a single current-carrying wire.