Maximum distance of charge from current carrying wire

[Algren]

Lets have a charged particle at a distance x at the beginning with velocity v away from a current carrying wire with current I. So, what will be the maximum distance of the particle from the wire? (only consider magnetic field)(wire is of infinite length)

There are two ways of solving the problem.

One is: Find forces in x and y-axis at a given time, get diff. equations, and solve em, and integrate tem.

Another is: We consider strips of magnetic field with 'dx' thickness, and integrate the deviation 'dθ' of the charged particle over all these strips from 0 to ∏/2. But in this case, i end up integrating sin(dθ). If i integrate sin(θ) dθ instead, i get the correct answer as so derived from the first way.

But i wanted to ask, is there any problem with the logic of the second way?

 

[sankalpmittal]

Another is: We consider strips of magnetic field with 'dx' thickness, and integrate the deviation 'dθ' of the charged particle over all these strips from 0 to ∏/2. But in this case, i end up integrating sin(dθ). If i integrate sin(θ) dθ instead, i get the correct answer as so derived from the first way.

But i wanted to ask, is there any problem with the logic of the second way?

How do you end up with integrating sin(dθ) ? You should be integrating sinθ w.r.t θ. Show me your work. You should have done any careless or conceptual mistake. Otherwise you shouldn't have arrived at such strange result.

Please post your second method here (with details and steps inclusive).