Magnetostatics Notes

Chapter 5: Magnetostatics

5.1 The Lorentz Force Law

  • Magnetic Fields: The Lorentz force law describes the interaction between charged particles and magnetic fields. In magnetostatics, we study the situations where charges are in motion, resulting in magnetic forces and fields.
    • Source Charges: These are the charges (q1, q2, q3, …) creating an electric or magnetic field.
    • Test Charge: Denoted as Q, it experiences forces due to the magnetic fields generated by source charges.
  • Demonstration of Magnetic Forces: When current flows in two parallel wires, forces between them can be observed.
  • Key Observations:
    • Parallel currents attract and antiparallel currents repel each other. This is evidence of the magnetic field generated by moving charges.

5.1.1 Magnetic Forces

  • Lorentz Force: The magnetic force acting on a charge Q with velocity v in a magnetic field B is given by the equation:
    • ( F_{mag} = Q(v \times B) )
    • In terms of electric and magnetic fields, the total force is ( F = Q[E + (v \times B)] ).
  • Motion in a Magnetic Field:
    • Charged particles in a uniform magnetic field exhibit circular motion. The centripetal force required for this motion is provided by the magnetic force.
  • Cyclotron Motion: Describes how a particle with charge Q and mass m moving in a magnetic field undergoes circular motion.
  • Helical Motion: A component of the particle's velocity aligned with the magnetic field remains unaffected, leading to a helical path.

5.1.2 Currents

  • Current Definition: Current (I) refers to the rate of flow of charge per unit time, measured in amperes (A), where 1 A = 1 C/s.
  • Vector Nature of Current: Currents are vectors, represented as ( I = \lambda v ), where ( \lambda ) is charge density, and v is the velocity vector of the charge.

5.1.3 Conclusion: No Work Done by Magnetic Forces

  • Magnetic forces do not do work; they only change the direction of the motion of charges. Work is done by the electric field or external agents maintaining current.

5.2 The Biot-Savart Law

  • Steady Currents: The Biot-Savart law relates steady currents (I) to the resultant magnetic fields (B).
  • The magnetic field generated by a steady current is given by:
    • ( B(r) = \frac{\mu_0}{4\pi} \int \frac{I \hat{r'}}{r^2}dl' )

5.2.1 Example Applications

  • Long Straight Wire: The magnetic field at a distance from a long straight wire carrying current I is:
    • ( B = \frac{\mu_0 I}{2\pi s} \hat{\phi} )

5.2.2 Ampère's Law

  • Ampère's Law: Relates the integral of the magnetic field B around a closed loop to the total current I encircled by the loop:
    • ( \int B \cdot dl = \mu0 I{enc} )

5.3 Divergence and Curl of B

  • The divergence of B is always 0 (no magnetic monopoles exist).
  • The curl of B relates directly to current density: ( \nabla \times B = \mu_0 J ).

5.4 Magnetic Vector Potential

  • Magnetic Vector Potential: Defined as ( B = \nabla \times A ). Changing this potential does not affect the magnetic field due to redundancy.

  • Boundary Conditions: The tangential components of the magnetic field B can be discontinuous across surfaces with surface currents, while the vector potential A remains continuous across such boundaries.

Summary of Key Equations

  • Lorentz Force: ( F_{mag} = Q(v \times B) )
  • Biot-Savart Law: ( B(r) = \frac{\mu_0}{4\pi} \int \frac{I \hat{r'}}{r^2}dl' )
  • Ampère's Law: ( \int B \cdot dl = \mu0 I{enc} )
  • Curl of B: ( \nabla \times B = \mu_0 J )
  • Divergence of B: ( \nabla \cdot B = 0 )
  • Magnetic Vector Potential: ( B = \nabla \times A )