EE3310 Lecture 12: The Biot-Savart Law

Key Concepts

• Magnetostatics: The study of magnetic fields produced by steady (DC) currents.
• Biosavar Law: A fundamental law that calculates the magnetic field intensity (H) at a point due to a differential current element.
• Differential Current Element (Idl): A small segment of current-carrying wire, where I is the current and dl is the differential length vector.
• Magnetic Field Intensity (H): A vector field that describes the magnetic influence of electric currents or magnetic materials.
• Magnetic Flux Density (B): A vector field related to H by the permeability (μ) of the medium (B = μH).
• Filamentary Line Current: A current flowing through an infinitesimally thin wire.
• Surface Current Density (K): Current flowing per unit width on a surface (Amps/meter).
• Volume Current Density (J): Current flowing per unit area within a volume (Amps/meter squared).
• Cylindrical Coordinates (ρ, φ, z): A coordinate system used to describe points in space using a radial distance (ρ), an azimuthal angle (φ), and a height (z).
• Cross Product: A vector operation that produces a vector perpendicular to the two input vectors, with a magnitude equal to the product of their magnitudes and the sine of the angle between them.

Comparison of Electrostatics and Magnetostatics

• Electrostatics: Deals with electric fields (E) created by stationary charges. The key equations are ∇ x E = 0 (curl of E is zero) and ∇ ⋅ D = ρv (divergence of electric flux density D equals volume charge density ρv).
• Magnetostatics: Deals with magnetic fields (H) created by moving charges (DC currents). The key equations are ∇ x H = J (curl of H equals current density J) and ∇ ⋅ B = 0 (divergence of magnetic flux density B is zero).
• Decoupling: In the time-invariant case, electric and magnetic fields are independent of each other.

Biosavar Law: Calculating Magnetic Fields

• The Law: The differential magnetic field intensity (dH) at a point P due to a differential current element (Idl) is given by:
  • dH = (Idl x r̂) / (4πr²)
  • Where:
    • Idl is the differential current element (current times differential length vector).
    • r̂ is the unit vector pointing from the current element to the point P.
    • r is the distance between the current element and the point P.
• Direction: The direction of dH is determined by the cross product of Idl and r̂, following the right-hand rule.
• Generalizations:
  • Line Current: H = ∫ (Idl x r̂) / (4πr²) (line integral along the current path)
  • Surface Current: H = ∫ (K x r̂) / (4πr²) dS (surface integral over the current-carrying surface)
  • Volume Current: H = ∫ (J x r̂) / (4πr²) dV (volume integral over the current-carrying volume)

Example 1: Magnetic Field Due to a Finite Line Current

• Problem: Calculate the magnetic field (H) at a point due to a straight, finite-length wire carrying a DC current.
• Setup:
  • The wire is aligned along the z-axis, extending from point A to point B.
  • Cylindrical coordinates (ρ, φ, z) are used due to the cylindrical symmetry of the problem.
  • The observation point P is defined by (ρ, φ, z).
• Steps:
  1. Define dl: dl = dz ẑ (differential length along the z-axis).
  2. Define r: r = ρ ρ̂ - z ẑ (vector from the current element to the observation point).
  3. Calculate dl x r: dl x r = ρ dz φ̂.
  4. Apply Biosavar Law: dH = (I ρ dz φ̂) / (4π (ρ² + z²)^(3/2)).
  5. Integrate: Integrate dH along the length of the wire (from z=a to z=b) to find the total magnetic field H. A trigonometric substitution (z = ρ cot(α)) is used to simplify the integral.
• Result: H = (I / (4πρ)) (cos(α₂) - cos(α₁)) φ̂, where α₁ and α₂ are the angles between the observation point and the start and end points of the wire, respectively.
• Key Observation: The magnetic field is in the φ̂ direction (azimuthal direction) and its magnitude is inversely proportional to the distance from the wire (1/ρ).

Example 2: Magnetic Field Due to a Circular Loop Current

• Problem: Calculate the magnetic field (H) at a point along the axis of a circular loop carrying a DC current.
• Setup:
  • The loop lies in the x-y plane with radius ρ.
  • The observation point P is on the z-axis at a height h.
  • Cylindrical coordinates (ρ, φ, z) are used.
• Steps:
  1. Define dl: dl = ρ dφ φ̂ (differential length along the loop).
  2. Define r: r = h ẑ - ρ ρ̂ (vector from the current element to the observation point).
  3. Calculate dl x r: dl x r = ρh dφ ρ̂ + ρ² dφ ẑ.
  4. Apply Biosavar Law: dH = (I (ρh dφ ρ̂ + ρ² dφ ẑ)) / (4π (ρ² + h²)^(3/2)).
  5. Integrate: Integrate dH around the loop (from φ=0 to φ=2π). Due to symmetry, the ρ̂ component cancels out. Only the ẑ component contributes to the total magnetic field.
• Result: H = (I ρ² / (2 (ρ² + h²)^(3/2))) ẑ.
• Key Observations:
  • The magnetic field is only in the ẑ direction (along the axis of the loop).
  • The magnitude of the field decreases as the distance from the loop increases.
  • Field lines form loops surrounding the current element.

Synthesis/Conclusion

The lecture provides a detailed introduction to magnetostatics, focusing on the Biosavar Law and its application to calculate magnetic fields generated by DC currents. The law is presented in its differential form and then generalized for line, surface, and volume current distributions. Two practical examples, the finite line current and the circular loop current, demonstrate the step-by-step process of applying the Biosavar Law, including the importance of choosing the appropriate coordinate system (cylindrical in these cases), defining the differential length element (dl) and the vector from the source to the field point (r), calculating the cross product (dl x r), and performing the necessary integration. The lecture emphasizes the vector nature of the magnetic field and the role of symmetry in simplifying calculations. The final results provide insights into the spatial distribution of magnetic fields around these common current configurations.