Magnetostatics Overview

• Magnetostatics involves the study of magnetic fields in systems where the currents are steady (not changing with time).

Biot-Savart Law

• Definition: The Biot-Savart Law states that at any point P, the magnitude of the magnetic field intensity (
  $ext{H}$) produced by a differential current element is proportional to:
    - The product of the current ($I$)
    - The magnitude of the differential length element ($dL$)
    - The sine of the angle ($heta$) between the current filament and the line connecting the filament to point P.
• Mathematical Expression: The magnetic field intensity $ext{H}$ is given by:
    - $ext{H} = \frac{1}{4 ext{π}} imes \frac{I imes dL imes ext{sin}( heta)}{R^2}$
       where $R$ is the distance from the filament to the point P where the field is calculated.
• Direction of Magnetic Field: The direction of $ext{H}$ is normal to the plane containing the differential filament and the line drawn from the filament to point P.

Vector Formulation

• The Biot-Savart law can be expressed in vector notation as:
     - $d ext{H} = \frac{I}{4 ext{π}} \frac{d ext{L} imes ext{R}}{R^3}$
       where $ext{R}$ is the position vector from the current element to the point P.

Comparison to Coulomb's Law

• The Biot-Savart law shows similarities to Coulomb’s law, particularly:
    - Both exhibit inverse-square law dependence on distance.
    - Both have a linear relationship between the source and the field.
• Mathematical Expression of Coulomb's Law:
     - $dE = \frac{1}{4 ext{π} ext{ε}} \frac{dQ imes a}{R^2}$
       where $dE$ is the electric field due to the charge element $dQ$ and $a$ is the unit vector in the direction of the field.

Application of the Biot-Savart Law

• Consideration of an Infinitely Long Straight Filament:
    - Illustrating the setup where the differential length $dL$ contributes to the field.
  

Establishing ${ ext{H}}$ for Straight Filament

• The setup assumes the infinite filament along the z-axis, the contributions from differential lengths $dz$ and calculations of $ext{H}$ throughout.
• To calculate $ext{H}$:
    - Analysis continues in terms of simplified components and contributions from the infinite extent of a straight filament.

Ampere’s Circuital Law

• Definition: Ampere’s circuital law states that the line integral of the magnetic field intensity $ext{H}$ around any closed path is equal to the current enclosed by that path.
• Mathematical Expression:
     - $ext{I}_{enc} = ext{∮ H} \bullet dL$

Application of Ampere’s Law to Magnetic Field Intensity

• Example Calculation: For an infinitely long filament carrying a current $I$:
     - The resulting magnetic field intensity can be derived:
     - $ext{H} = \frac{I}{2 ext{π} ext{R}}$
• Exploring different regions:
    - Inside and outside considerations of coaxial conductors and circular paths of different radii.
       - $ext{H} = \frac{I_{encl}}{2 ext{π} ext{R}}$ for appropriate path selections.

Curl and Divergence

• Concept of Curl: Derived from applying Ampere’s law over differential surfaces to elucidate current density variations in magnetic fields.
• $ext{Curl H}$ signifies how field intensity varies in space and helps isolate magnetic scenarios.

Mathematical Representation of Curl

• Expressed in Cartesian coordinates:
    - ext{Curl H} = egin{pmatrix} rac{ extrm{d} H_z}{ extrm{d} y} - rac{ extrm{d} H_y}{ extrm{d} z} \ rac{ extrm{d} H_x}{ extrm{d} z} - rac{ extrm{d} H_z}{ extrm{d} x} \ rac{ extrm{d} H_y}{ extrm{d} x} - rac{ extrm{d} H_x}{ extrm{d} y} \
   ext{end{pmatrix}}.

Maxwell's Equations in Magnetic Context

• Maxwell’s Second & Third Equations:
    -
  abla imes ext{H} = ext{J}
    -
  abla imes ext{E} = 0
• Interpretations of these equations provide essential context for electromagnetic theory in static fields, aiding in understanding field behaviors.

Magnetic Flux and Density

• Definition: Magnetic flux density is defined under free-space conditions as:
    - ext{B} = ext{μ}_0 ext{H}
    - Where ext{μ}_0 is the permeability of free space:
     - ext{μ}_0 = 4 ext{π} imes 10^{-7} rac{N}{A^2}
• Gauss's Law for Magnetism: States that the total magnetic flux through a closed surface is equal to zero:
    -
  abla ullet ext{B} = 0$$

Inductance

• Definitions:
    - Inductance (Self Inductance): The ratio of magnetic flux linkages to the current providing them.
    - Mutual Inductance between two circuits relates the flux in one circuit to the current in another.
• Units: The basic unit for measuring inductance is the Henry (H).