There exists a uniform magnetic field `vec(B) = +B_(0) hat(k) " for " x gt 0 and vec(B) = 0` for all `x lt 0`. A charged particle placed at `(0, -a,0)` is given an initial velocity `vec(v) = v_(0) hat(i)`. What is the magnitude and nature of charge on the particle such that it crosses through the origin ?
Strategy: When the charge is projected perpendicular to a uniform magnetic field, it follows a circular path. In this case, the force acting on it will be directed either towards +y-axis. or y-axis. It is given that it crosses point O. Thus, the force at `(0, -a, 0)` must be towards y-axis

Explanation

To solve the problem, we will analyze the motion of a charged particle in a magnetic field and determine the nature and magnitude of the charge.

### Step-by-Step Solution:

1. **Understanding the Magnetic Field**:
The magnetic field is given as:
\[
\vec{B} = B_0 \hat{k} \quad \text{for } x > 0
...