SUMMARY
The conservation of the quantity $$\vec J=\vec r \times\vec p +eg\frac {\vec r}{|\vec r|}$$ in a magnetic field defined by $$B(\vec r)=g \frac {\vec r}{|\vec r|^3}$$ indicates that the motion of an electron is constrained to the surface of a cone. By manipulating the equation $$\vec{r} \cdot \vec{J}=e g r$$ and applying spherical coordinates, it is established that the angle $$\vartheta$$ remains constant, leading to the conclusion that the trajectory forms a cone. The relationship $$\cos \vartheta=\frac{e g}{J}$$ confirms the constancy of the angle, solidifying the conical motion.
PREREQUISITES
- Understanding of classical mechanics, specifically angular momentum.
- Familiarity with magnetic fields and their effects on charged particles.
- Knowledge of spherical coordinates and their application in physics.
- Proficiency in vector calculus, particularly operations involving cross products.
NEXT STEPS
- Explore the principles of angular momentum conservation in electromagnetic fields.
- Study the mathematical representation of magnetic fields, focusing on vector fields.
- Investigate the implications of conical motion in classical mechanics.
- Learn about the applications of spherical coordinates in physics problems.
USEFUL FOR
Students of physics, particularly those studying electromagnetism and classical mechanics, as well as educators seeking to explain the concepts of motion in magnetic fields.