Magnetic Fields, Magnetostatics, Biot-Savart Law

• Magnetic fields are vector fields, denoted as B, created by moving charges.
• The magnetic field due to a moving point charge is given by: B = (μ₀ / 4π) * (qν × R) / |R|².
• Magnetostatics and the Biot-Savart law will be covered.
• Distance Vector: R = r - r', where R is the physical distance vector.
  • r is the vector from the origin to the point where the field is calculated.
  • r' is the vector from the origin to the source of the field.
  • R is the vector from the source to the point of calculation.
• Magnetic fields due to moving point charges curl around the charge.

Magnetostatics

• Study of magnetic fields from constant (steady) current sources, similar to electrostatics with stationary charges.
• Constant current I creates a magnetic field B, the sum of contributions from moving charges via superposition: Btot = B1 + B2 + B3 + ...

Biot-Savart Law

• Gives the magnetic field dB due to a current element: dB = (μ₀ / 4π) * (Idl × R) / |R|².
• sin definition: dB = (μ₀ / 4π) * (Idl sin φ) / R², where R is the distance from the wire element to the point P.
• For an extended wire, integrate the equation to find the total magnetic field.

Magnetic Field due to a Wire of Length 2a

• Formula: B = (μ₀ / 4π) ∫ (Idl × R) / |R|²
• Considering wire of length 2a on the y-axis:
  • r = <x, 0, 0>
  • r' = <0, y, 0>
  • R = <x, -y, 0>
  • |R| = √(x² + y²)
  • Unit vector R = (1/√(x² + y²)) <x, -y, 0>
• dl is in the direction of current: dl = <0, dy, 0>
  • dl = |dl|Î, where |dl| = dy and Î = <0, 1, 0>
• dB = (μ₀ / 4π) * (Idl × R) / |R|² can be calculated by finding dl x R
  • dl × R = <0, 0, xdy / √(x² + y²)
  • dB = -((μ₀ * I) / (4π)) * (x dy) / (x² + y²)^(3/2) k
• Integrating from y = -a to y = a:
  • x is constant wrt integration
  • B = -k (μ₀ * I) / (4π) * [y / (x²√(x² + y²)] from -a to a

Magnetic Field due to Infinite Wire

• derived from previous equation, when y goes to infinity
• B = -(k μ₀ I) / (4π) * (2a / (x √(x² + a²))
• B = -(k μ₀ I) / (2πx)
• Magnitude: B = (μ₀I) / (2πr), where r is the distance from the wire.
• Standard introductory result for magnetostatics.