(a) A nonrelativstic particle with charge q and mass m moves in a constant field B...

 

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(a) A nonrelativstic particle with charge q and mass m moves in a constant field B = B,2 , and Ē = 0. Determine the parameters of the resulting helical trajectory in terms of the initial velocity components v, (0) and v (0) parallel and perpendicular to B. In particular answer the following: 1) What is the radius of the resulting helical motion? 2) How far along the z direction does a particle advance during one turn? 3) What is the magnetic moment u of the system, assuming the motion in the z direction is very slow? (b) Now B is given a spatial dependence of the form: B (7, , 2)=B_(r. Dî +(B, +az)î (3) where r=(x, y,0). 1) Use the condition that there are no magnetic monopoles to calculate B ( D. dv. in terms of a and . 2) Derive an equation for dt 3) Using these results can you suggest a way of confining particles in a “magnetic bottle"? Sketch qualitatively the spatial distribution of B. required for confinement. (a) A nonrelativstic particle with charge q and mass m moves in a constant field B = B,2 , and Ē = 0. Determine the parameters of the resulting helical trajectory in terms of the initial velocity components v, (0) and v (0) parallel and perpendicular to B. In particular answer the following: 1) What is the radius of the resulting helical motion? 2) How far along the z direction does a particle advance during one turn? 3) What is the magnetic moment u of the system, assuming the motion in the z direction is very slow? (b) Now B is given a spatial dependence of the form: B (7, , 2)=B_(r. Dî +(B, +az)î (3) where r=(x, y,0). 1) Use the condition that there are no magnetic monopoles to calculate B ( D. dv. in terms of a and . 2) Derive an equation for dt 3) Using these results can you suggest a way of confining particles in a “magnetic bottle"? Sketch qualitatively the spatial distribution of B. required for confinement.