Solution
Problem Analysis:
A particle of charge $q$ and mass $m$ is projected from point O with velocity $u$ at an angle $\theta$ with the magnetic field $B$. The particle performs helical motion. We need to find the value of $B$ such that the particle passes through point P, which is at a distance $l$ from O along the direction of the magnetic field.
Step 1: Analyze Motion Components
The velocity component parallel to the field is $u_{\parallel} = u \cos\theta$.
The velocity component perpendicular to the field is $u_{\perp} = u \sin\theta$.
The particle moves in a helix. The time period of one revolution is given by:
$$ T = \frac{2\pi m}{qB} $$Step 2: Pitch of the Helix
The pitch ($p$) is the linear distance covered along the magnetic field in one time period:
$$ p = u_{\parallel} \times T = (u \cos\theta) \frac{2\pi m}{qB} $$Step 3: Condition for Intersecting Point P
For the particle to pass through point P at distance $l$, the distance $l$ must be an integral multiple of the pitch. That is, the particle must complete $n$ full revolutions to return to the axis at distance $l$.
$$ l = n \times p \quad \text{where } n = 1, 2, 3… $$ $$ l = n \left( \frac{2\pi m u \cos\theta}{qB} \right) $$Solving for the magnetic induction $B$: