Moving Charges and Magnetism - Lecture 2

Jul 24, 2024

Moving Charges and Magnetism - Lecture 2 Summary

Introduction

  • Discussion on magnetic fields generated by current-carrying wires.
  • Recap on Biot-Savart Law from Lecture 1:
    • Current-carrying wires create magnetic fields.
    • Biot-Savart Law provides a way to calculate the magnetic field produced by such wires.

Key Concepts

Biot-Savart Law

  • Formula:
    • Scalar form:
      [ dB = \frac{\mu_0}{4\pi} \frac{I , dL , \sin(\theta)}{R^2} ]
      • B: Magnetic field
      • dL: Length of the wire segment in the direction of current
      • I: Current
      • R: Distance from the current element to the point of interest
      • ( \theta ): Angle between the wire segment and the line connecting the segment to the point
  • Vector form: Used to calculate the direction of dB.

Direction of Magnetic Field

  • Right-hand rule to determine direction of magnetic field:
    • Thumb in current direction, fingers in field direction.

Magnetic Field from Circular Current-Carrying Loop

  • For a circular loop with a radius a,
    • At the center, magnetic field is given by: [ B = \frac{\mu_0}{2} \frac{I}{a} ]
  • Direction: Inward if current is anti-clockwise, outward if clockwise.

Magnetic Field from Semi-Circular Wire

  • For a semi-circular wire, utilize the same Biot-Savart Law:
    • Magnitude: [ B = \frac{\mu_0}{4} \frac{I}{a} ]
  • Direction: Inward for clockwise current.

Quadrant Wire

  • For a quadrant of a circle:
    • Magnetic field at the center: [ B = \frac{\mu_0}{8} \frac{I}{a} ]

Magnetic Field of General Arc

  • For an arc with an angle θ, the magnetic field is:
    • At the center: [ B = \frac{\mu_0}{4\pi} \frac{I \theta}{a} ]
  • Angle is in radians.

Magnetic Field Due to Straight Wire

  • If a point is along a straight wire, the magnetic field due to that wire is zero.

Combined Systems

Case with Multiple Wires

  • For systems where multiple currents flow and intersect:
    • Analyze each segment's contribution using Biot-Savart Law.
    • Consider the resistance in the circuit when calculating resultant magnetic fields.

Important Takeaways

  • Review derivations with focus on magnitudes and directions of magnetic fields from different geometries:
    • Infinite wires, circular loops, semi-circles, etc.
  • Practice conceptual problems based on the calculations and concepts discussed in this lecture.