Regle Yüzeyin Manyetik Merkez Normali

Abstract

Bir regle yüzeyin daralma noktasındaki birim normalin yönüne, regle yüzeyin merkez normali denir ve $t$ ile gösterilir. Bu çalışmada, manyetik alan içindeki bir regle yüzeyin merkez normalinin Frenet çatısı $\left\{ T_{t}, N_{t}, B_{t}\right\}$ tarafından indüklenen Lorentz kuvvetleri ve manyetik eğrileri incelenmektedir. $T_{t}$, $N_{t}$, ve $B_{t}$ için manyetik eğrilere karşılık gelen manyetik vektör alanları belirlenmiştir. Ayrıca, bu manyetik vektör alanları tarafından manyetik merkez normali boyunca oluşturulan manyetik akı yüzeyleri kavramını tanıtılmış ve geliştirilebilirlik koşulları türetilmiştir. Son olarak, manyetik akı yüzeylerinin yapısını göstermek için örnekler sunulmuştur.

Keywords

• manyetik eğri
• merkez normal
• regle yüzey

Magnetic Central Normal of a Ruled Surface

Abstract

The direction of the unit normal at a striction point is called the central normal of a ruled surface and denoted by $t$. In this study, we examine the Lorentz forces and magnetic curves induced by the Frenet frame $\left\{ T_{t}, N_{t}, B_{t}\right\}$ of the central normal of a ruled surface within a magnetic field. We determine the magnetic vector fields corresponding to the magnetic curves for $T_{t}$, $N_{t}$, {and} $B_{t}$. Furthermore, we introduce the concept of magnetic flux surfaces generated by these magnetic vector fields along the magnetic central normal and derive their developability conditions. Finally, we present examples to illustrate the structure of the magnetic flux surfaces.

Keywords

• magnetic curve
• central normal
• ruled surface

References

1. Barros, M., & Romero, A. (2007). Magnetic vortices. Europhysics Letters, 77(3), 34002.
2. Barros, M., Romero, A., Cabrerizo, J. L., & Fernandez, M. (2005). The Gauss-Landau-Hall problem on Riemannian surfaces. Journal of Mathematical Physics, 46(11).
3. Chen, B. Y., Aloui, F., Khan, M. A., & Choudhary, M. A. (2025). Magnetic curves in differential geometry: a comprehensive survey. Mathematics, 13(23), 3849.
4. Xu, L., & Mould, D. (2009). Magnetic curves: curvature-controlled aesthetic curves using magnetic fields. The Eurographics Workshop on Computational Aesthetics in Graphics, Visualization, and Imaging (CAe 2009), Canada, (p. 1--8).
5. Hazeltine, R. D., & Meiss, J. D. (2003). Plasma confinement. Courier Corporation.
6. Bird, R. B. (2002). Transport phenomena. Applied Mechanics Reviews, 55(1), R1--R4.
7. Boozer, A. H. (2004). Physics of magnetically confined plasmas. Reviews of Modern Physics, 76(4), 1071--1141.
8. Pedersen, T. S., & Boozer, A. H. (2002). Confinement of nonneutral plasmas on magnetic surfaces. Physical Review Letters, 88(20), 205002.

9. Güler, F., Bayram, E., & Kasap, E. (2023). Magnetic spherical indicatricies in Minkowski 3-space. International Journal of Geometric Methods in Modern Physics, 20(11), 2350179.
10. Frenet, J. F. (1852). Sur les courbes à double courbure. Journal de Mathématiques Pures et Appliquées, 17, 437--447.
11. Serret, J. A. (1851). Sur quelques formules relatives à la théorie des courbes à double courbure. Journal de Mathématiques Pures et Appliquées, 16, 193--207.
12. Arnold, V. I., & Khesin, B. A. (2009). Topological methods in hydrodynamics. Springer, New York.
13. Vogel, M. (2024). Particle confinement in penning traps. Springer Series on Atomic, Optical, and Plasma Physics, 126, Switzerland.
14. Ravani, B., & Ku, T. S. (1991). Bertrand offsets of ruled and developable surfaces. Computer-Aided Design, 23(2), 145--152.
15. Somasundaram, D. (2005). Differential geometry: a first course. Alpha Science Int'l Ltd.
16. O'Neill, B. (2006). Elementary differential geometry. Elsevier.
17. Yıldırım, A., & Kasap, E. (2024). Magnetic spherical indicatrix of a ruled surface. International Journal of Geometric Methods in Modern Physics, 21(03), 2450068.