Chapter 15: Vector Potential & Magnetic Energy

Vector Potential & Magnetic Energy physics chapter begins by establishing how a current loop acts as a magnetic dipole, deriving the standard formulas for the rotational force (torque) it experiences in a uniform magnetic field and the related mechanical potential energy. The text then clarifies a critical energy concept by distinguishing between the mechanical work done and the true total energy stored in steady currents. For systems with constant current, the true total energy of a magnetic dipole in a field is the negative of the mechanical energy. This foundational understanding allows for the development of the general formula for the energy of steady currents, expressed as an integral over the volume where the current density exists, using the magnetic vector potential. The chapter transitions into a profound philosophical and physical debate: whether the magnetic field (B) or the vector potential (A) is the genuinely physical field. While classical physics favors B via the Lorentz force, the discussion turns to quantum mechanics to demonstrate the physical significance of the vector potential A. Analyzing electron interference, the text shows that the vector potential, through its line integral, determines the magnetic change in the phase of a quantum particle's wave function. The significant Aharonov-Bohm effect is presented as proof, showing that a concentrated magnetic field inside a solenoid can still affect electrons traveling in regions where the magnetic field B is effectively zero. This confirms that the observable quantum phase shift fundamentally relies on the vector potential A. Finally, the chapter concludes with a table summarizing the complete electromagnetic equations (Maxwell's full theory), distinguishing them from the approximation formulas valid only for magnetostatics.