CHPTR 28 Notes: Sources 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).

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.

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}

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.

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}1 + extbf{B}2

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{B12} = rac{μ0 I1 I2}{2 ext{π}d}
- Here, I1 and I2 are the currents in each wire and d is the distance between the wires.
Another perspective gives:
FB = B2 I_1 L ext{sin}θ where L is the length of the wire in the other wire's magnetic field.

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

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}}

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.

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} ullet d extbf{s} = μ0 I{enc}

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}

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.

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.

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.

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}

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.

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.