应用欧拉φ函数研究带电粒子在磁场中的一类特殊运动——基于2023年高考全国理综甲卷第20题的深度思考

Applying EulerφStudy on a Special Motion of Charged Particles in a Magnetic Field Using Functions——Deep Reflection on Question 20 of the 2023 National College Entrance Examination Science Comprehensive Examination Volume A

深度研究2023年高考全国理综甲卷第20题,给出带电粒子能够穿出圆筒的条件;应用欧拉φ函数对带电粒子运动轨迹进行了深入研究,给出粒子运动的可能轨迹数目确定表达式,对轨迹形状予以详细讨论;分析了带电粒子在磁场中运动的总时间问题,给出这一总时间跟粒子与圆筒碰撞次数和粒子在圆筒中绕圆筒轴心运动的圈数之间的约束关系.

Conduct in-depth research on Question 20 of the 2023 National College Entrance Examination Science Comprehensive Volume A,and provide the conditions for charged particles to pass through the cylinder;Applying EulerφThe function has conducted in-depth research on the motion trajectory of charged particles,provided an expression for determining the number of possible trajectories of particle motion,and discussed the trajectory shape in detail;The total time problem of charged particles moving in a magnetic field was analyzed,and the constraint relationship between this total time and the number of collisions between particles and the cylinder,as well as the number of circles of particles moving around the cylinder axis in the cylinder,was given.