A positively charged particlewith charge "q" is at $ (0,0,0)$.There is a uniform magnetic field $ \vec B= -B \hat k$. The particle is given velocity $\vec u= u \hat i$. A resistive friction acts on it which is proportional to is velocity $(\vec F_{friction} = - k \vec v)$. Find the equation of the trajectory of its motion on the trajectory of its motion

I considered its velocity at any time t to be $$\vec v = v_x \hat i + v_y \hat j$$ Now Force on it is $$\vec F= q(v_x \hat i + v_y \hat j)×(-B \hat k) - k( v_x \hat i + v_y \hat j)$$

So $$\vec F= q(-v_y B - k v_x) \hat i+ q(v_x B - k v_y) \hat j$$

So know I get 2 differential equations $$ m\frac{d V_x}{dt} = q(-v_y B - k v_x )$$ And $$ \frac{dV_y}{dt} = q(v_x B - k v_y )$$

Now what... How to solve these ??

[ What I originally planned was to seperate these differential equations to get velocity of $ v_x and v_y $ and then integrate it {Like we do it in the question with cycloid, where there is an $\vec E$ field perpendicular to $\vec B$ field }]

(I'm in highschool so , maybe this might be out of my reach buy since I was presented this problem, I assume there surely must be some clever trick to solve it, or any other shorter approach. Please help :)


You should seek a solution of the form $v_x = A\exp(l\cdot t)$ and $v_y = B\exp(l\cdot t)$. After you put it into the equations, you will received two linear algebraic equations for $A$ and $B$ ($l$ is a parameter). Choose a parameter so that there are non-trivial solutions for $A,\,B$. And solve this two linear equations with the parameter $l$ (maybe there will be two suitable values for $l$ and you should solve with each one separately).

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