Particle in a mag field with linear drag

[Liquidxlax]

Homework Statement

A particle of mass m and electric charge q>0 is subject to a uniform, constant mag field of $$\vec{B} = B \hat{k}$$. At t=0 its velocity vector is $$\hat{v}$$naught which lies in the x-y plane. The particle is also subject to a linear drag force f_d=-b$$\vec{v}$$

a)Draw a free body diagram showing all the forces action 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 (see also taylor, section 2.5) to solve for the complex velocity n(t)= v_x+iv_y as a function of time, in terms of the initial condition $$\vec{v}$$naught

c) use your solution to part b) to find an expression for the particle's speed as a function of time.

hate that my prof links the questions...

Homework Equations

$$\omega = \frac{qB}{m}$$

a_y = -$$\omega v_{x}$$

a_x = $$\omega v_{y}$$

-bv = $$\beta Dv$$

The Attempt at a Solution

Lorentz force law where E=0 qvxb = qB{i j k}{v_x v_y 0}{0 0 k}

=qB(v_yi - v_xj)


F = ma = qB(v_yi - v_xj) - b(v_yi - v_xj)

after some shuffling around dividing by mass subbing in a_y and a_x

[1-(b/m$$\omega$$)]*[ a_y + a_x ] = a

multiply by mass

F= ma = [m - (b/$$\omega$$)]* [ a_y + a_x ]

I'm not sure if i was supposed to make this a function of t or if that is with part b

 

[Liquidxlax]

anyone have an idea?

 

[Liquidxlax]

ehild your a smart guy. want to help a guy out?

 

[Liquidxlax]

I figured it out, needed to clear my head and it was easy

 

[rwisz]

b) I'm not sure how to use the complex variable technique to solve for the complex velocity. It would involve using the complex form of the equations of motion and solving for the complex variables vx and vy. This can be done by taking the real and imaginary parts of the equations and solving for each variable separately. The solution would involve the initial conditions and the exponential function e^(bt/m) where b is the drag coefficient and t is time.

c) To find the particle's speed as a function of time, we can use the solution from part b and take the magnitude of the complex velocity. This would involve using the Pythagorean theorem to find the magnitude of the complex velocity, which would give us the speed as a function of time.