Lecture Notes on Biot-Savart Law and Applications

Introduction

• Biot-Savart Law: Fundamental in determining the magnetic field produced by a current-carrying conductor.
• Empirical observations show the magnetic field dB at a point P is proportional to:
  • Current (I)
  • Length element (dl)
  • Inversely proportional to the distance squared (1/r^2)
  • Dependent on the angle between dl and r (sin θ)

Formula Derivation

• Biot-Savart Law:
  • dB ∝ (I * dl * sin θ) / r^2
  • In SI units: dB = (μ₀/4π) * (I * dl * sin θ) / r²*

Applications

Magnetic Field Due to a Circular Coil

• At the center:
  • Magnetic field B is derived from summing contributions of all small segments of the coil.
  • B = (μ₀ * I * r²) / (2 * (r² + x²)^(3/2))
  • For a coil with multiple turns (n): B = (μ₀ * n * I * r²) / (2 * (r² + x²)^(3/2))
• Special Cases:
  • At center (x = 0), simplifies to: B = (μ₀ * I * n) / (2 * r)

Magnetic Field of a Straight Conductor

• Uses integration to find magnetic field at a point P due to the entire length of a conductor.
• Derivation involves trigonometric relations and integration techniques.
• Special case for infinitely long wires results in a simpler expression for B.

Magnetic Field on the Axis of a Circular Coil

• Results in B = (μ₀ * I * R²) / (2 * (R² + x²)^(3/2))
• If point P is at the center of the coil, simplifies to the same formula for a coil's center.*

Magnetic Field Inside a Solenoid

• Solenoid: A coil of wire often wrapped in a helix shape.
• Magnetic field due to a solenoid is calculated considering infinitesimal coil elements.
• Formula: B = μ₀ * n * I, where n is number of turns per unit length.

Helmholtz Coil Arrangement

• Helmholtz Coils: Used to create a uniform magnetic field in a region.
• Consists of two identical circular coils, separated by a specific distance for uniformity.
• Formula for field in the uniform region: B = (μ₀ * I * R²) / (2 * (R² + (R/2)²)^(3/2))*

Conclusion

• Biot-Savart Law is crucial for calculating magnetic fields from various geometries like straight wires, coils, and solenoids.
• Understanding and applying it allows for precise control and measurement of magnetic fields in physics and engineering.