CHPTR 28 Notes: Sources of Magnetic Fields

Source of Magnetic Fields

• Moving charges as well as time-varying electric fields are sources of magnetic fields.

• Focus of the study: Production of magnetic fields by moving charges.

• Introduction of the second right-hand rule, termed the “Hitchhiker” right-hand rule (RHR).

Moving Charge: Magnetic Field

• The magnetic field produced by a moving charge is expressed by the formula: $extbf{B} = rac{ extbf{r} imes q extbf{v}}{4 ext{π}\boldsymbol{ u}_{0}}$

  • Variables:

  • q = charge

  • v = velocity of charge

  • μ0 = permeability of free space, defined exactly as $4 ext{π} imes 10^{-7} ext{Tm/A}$.

  • ε0 = permittivity of free space, relevant to electric fields.

Biot-Savart Law

• The magnetic field ($extbf{dB}$) due to an elemental length of wire ($ds$) carrying current $I$ at point $P$ is: $extbf{dB} = rac{μ_0}{4π} rac{I extbf{ds} imes extbf{r}}{|r|^2}$ where $extbf{r}$ is the vector from $ds$ to $P$.

  • This formula helps in calculating the magnetic field contribution at a point from a small segment of current-carrying wire.

Total Magnetic Field

• The total magnetic field $extbf{B}$ at point $P$ due to a current-carrying conductor is given by the integral:
  $extbf{B} = rac{μ_0}{4π} imes ext{I} imes ext{∫} rac{ extbf{ds} imes extbf{r}}{|r|^2}$

Infinitely Long Straight Wire and the Second RHR

• The magnetic field lines due to infinitely long straight wires are circular and concentric with the wire.

• The magnitude of the magnetic field produced by an infinite wire is given by: $B = rac{μ_0 I}{2πr}$

  • Apply the second right-hand rule:

  • Thumb in the direction of the current ( extit{I}).

  • Curl fingers around to indicate the direction of the magnetic field surrounding the wire.

Multiple Current-Carrying Wires

• When two or more current-carrying wires are present, the magnetic field at any location is the vector sum of the fields from each wire:
  $extbf{B} = extbf{B}<em>1 + extbf{B}</em>2$

Magnetic Force Between Conductors

• Two parallel wires carrying steady currents exert forces on each other because each wire exists within the magnetic field created by the other wire.

• If the currents are parallel, the force is attractive; if the currents are anti-parallel, the force is repulsive.

• The formula for the magnitude of the force per unit length between the two wires is: $F<em>{B12} = rac{μ</em>0 I<em>1 I</em>2}{2 ext{π}d}$

  • Here, $I<em>1$ and $I</em>2$ are the currents in each wire and $d$ is the distance between the wires.

• Another perspective gives:
  $F<em>B = B</em>2 I_1 L ext{sin}θ$ where $L$ is the length of the wire in the other wire's magnetic field.

Current-Carrying Rings or Loops

• Comparison of the magnetic field lines surrounding a current loop with those around a bar magnet shows similarities.

Magnetic Field of a Coil or Loop

• The magnitude of the magnetic field at points along the central axis of a current-carrying loop with $N$ turns and radius $a$ is given by:
  $B(x) = rac{μ_0 N I a^2}{2(a^2 + x^2)^{3/2}}$

Magnetic Field at the Center of a Circular Loop

• The magnitude of the magnetic field at the center of a current-carrying loop of radius $a$ is given by: $B = rac{μ_0 N I}{2a}$

  • Again, $N$ is the number of loops or turns of wire.

Ampere’s Law

• The line integral of the magnetic field $extbf{B}$ around any closed loop is proportional to the total current passing through the surface bounded by that loop:
  $ext{∫} extbf{B} \bullet d extbf{s} = μ<em>0 I</em>{enc}$

Long Wire Application of Ampere’s Law

• For a long straight wire of radius $R$ carrying a current $I$ uniformly distributed, the magnetic field is given by:

  • For r < R:
    $B(r) = rac{μ_0 I r}{2π R^2}$

  • For r > R:
    $B(r) = rac{μ_0 I}{2π r}$

Solenoids

• In a tightly-wound solenoid, the magnetic field is uniform and strong, expressed as: $B = μ_0 n I$

  • Where $n$ is the number of turns per unit length, and $I$ is the current.

Toroids (Toroidal Solenoids)

• Using Ampere’s Law, the magnetic field inside a tightly-wound toroid is given by: $B = rac{μ_0 N I}{2π r}$

  • The magnetic field direction is tangent to the dashed circle.

Magnetic Effects from Orbiting Electrons

• In atoms, each electron orbits the nucleus once approximately every $10^{-16}$ seconds, producing a current of about $(1.6 ext{ mA})$ and a magnetic field of around $(20 ext{ T})$ at the circular path center.

• However, in many materials, oppositely moving electrons can nullify these magnetic effects, rendering them either zero or very small.

Electron Spin

• The magnetic moment associated with the spin of an electron is referred to as the Bohr magneton, expressed as:
  $μ_B = 9.27 imes 10^{-24} ext{ J/T}$

Atomic Moments

• In atoms with many electrons, electrons often pair up with opposite spins, leading to the cancellation of their magnetic moments.

• Atoms with an odd number of electrons generally will have non-vanishing magnetic moments.

• The overall magnetic moment of an atom results from the vector sum of both spin and orbital magnetic moments.

Magnetic Materials

• Magnetic substances can be classified as:

  • Ferromagnetic

  • Materials that have domains where all magnetic moments are aligned.

  • Paramagnetic

  • Materials with atoms that possess permanent magnetic moments.

  • Diamagnetic

  • Comprise atoms that exhibit no permanent magnetic moments.

Ferromagnetism

• Ferromagnetic materials show domains comprising approximately $(10^{-10} m^3)$ and around $10^{20}$ atoms.

• In an un-magnetized sample, domains are randomly oriented, but exposure to an external magnetic field can align these domains, resulting in permanent magnetization.