Particle in a Magnetic Field with Linear Drag A particle of mass m and electric
ID: 1602802 • Letter: P
Question
Particle in a Magnetic Field with Linear Drag A particle of mass m and electric charge q > 0 is subject to a uniform, constant magnetic field B = Bk. At t = 0, its velocity vector is v_0, which lies in the x - y plane. The particle is also subject to a linear drag force f_d = -bv. a) Draw a free body diagram showing all forces acting on the particle. Use this to find expressions for F = ma in the x and y directions respectively, assuming that the force of gravity is negligible. b) Use the equations from part a) and the complex variable technique discussed in class to solve for the complex velocity eta (t) = v_x + iv_y as a function of time, in terms of the initial condition v_o c) Use your solution to part b) to find an expression for the particle's speed as a function of time.Explanation / Answer
Ans:- As given mass =m, electric charge Q>0, B = constant magnetic field at time=0 in x-y plane
(a).so our free body diagram when a mass of object was reach in terminal velocity .so force of air Fair = Fgrav force of gravity. as per our condition force of gravity is negligible. Then the moment was opened made a fair even than greater than gravity force before. so we represented by the larger arrow. so that only this below diagram F=m*a is lies in x and y plan direction. it means vertical respectively. the force lies in x-y direction so that Fn - mg * cos if gravity is zero so we set is equal to 0 because f approx m*a = 0 so that path blocked accelerates in incline direction. Fy = Fn - mg cos = 0; so Fn = mg cos
(b). as given above equation Fn = mg cos . suppose that F(z) = -V0z where V0 is real.
(r,)= - V0 r cos
(r,)= - V0 r sin
As comparison with the analysis the complex variable techniques velocity potential n(t) = vx +ivy as corresponds to uniform function flow of speed Vo is this directed along with z -axis. Furthermore, the complex variable velocity potential is related to uniform flow of velocity Vo. that direction is reversible clockwise and angle with the z -axis
F(z) = -V0 z e -io.
(r ,theta) = - V0 r cos theta
(r ,theta) = - V0 r sin theta
suppose F(z) = - Q/2pielog z
if Q is complex variable function. Because log z = logr + i theta
(r ,theta) = - Q /2pie log r
(r ,) = - Q /2pie theta
Use the equations to solve the complex velocity function (t) = vx + ivy as time of function in the initial condition ~vo.
c). the particle’s speed as a function a Complex & Stream potensial function. so that Complex velocity factors together and define new variable names to simplify your equations
= qB/m
Vorticity: = x u , flow are not rottational
= x u = 0
scalar potential u = *thetaX= 0 , if flow are not compressible * u = 0,
× = 2= 0
complex velocity: dw / dz= u - iv. which = constant are streamlines so that complex velocity of particle’s speed as a function of time