A particle of specific charge `(q//m)` is projected from the origin of coordinate with initial velocity `[ui-vj]` unifrom electric magnetic fields exist in the region along the +ydirection, of magnitude `E` and `B` The particle will definitely return to the origin once if .

Options

A. `[vb//2piE]` is an integer
B. `(u^(2)+v^(2))^(1//2)[B//piE]` is an integer
C. `[vB//piE]` in an integer
D. `[uB//piE]` is an integer

Explanation

To solve the problem, we need to analyze the motion of a charged particle in the presence of uniform electric and magnetic fields. The particle is projected with an initial velocity and will follow a helical path due to the combined effects of the electric and magnetic fields. We will derive the condition under which the particle returns to the origin.

### Step-by-Step Solution:

1. **Understanding the Motion**:
The particle is projected from the origin with an initial velocity of \([u i - v j]\). The electric field \(E\) is in the positive \(y\)-direction, and the magnetic field \(B\) is also in the positive \(y\)-direction. The particle will experience a force due to both fields.

**Hint**: Identify the direction of the forces acting on the particle due to the electric and magnetic fields.

2. **Force on the Particle**:
The force \(F\) acting on the particle can be expressed using the Lorentz force equation:
\[ F = q(E + v \times B) \]
Since both \(E\) and \(B\) are in the \(y\)-direction, the magnetic force will act perpendicular to the velocity vector.

**Hint**: Recall that the magnetic force is given by the cross product and will affect the motion in the \(x\) and \(z\) directions.

3. **Helical Motion**:
The particle will follow a helical path due to the continuous influence of the magnetic field. The motion in the \(y\)-direction will be influenced by the electric field, while the motion in the \(x\) and \(z\) directions will be circular due to the magnetic field.

**Hint**: Visualize the helical path and how the particle moves in the \(y\)-direction while also revolving around the \(y\)-axis.

4. **Time Period of Motion**:
The time taken for the particle to complete one full revolution in the \(x-z\) plane is related to the magnetic field and the charge-to-mass ratio of the particle. The time period \(T\) can be expressed as:
\[ T = \frac{2\pi m}{qB} \]

**Hint**: Remember that the time period is dependent on the mass of the particle and the magnetic field strength.

5. **Condition for Returning to Origin**:
For the particle to return to the origin, the time taken to reach the \(x-z\) plane must be an integral multiple of the time period \(T\). This can be expressed mathematically as:
\[ t = nT \]
where \(n\) is an integer.

**Hint**: Relate the distance traveled in the \(y\)-direction to the electric field and the time taken.

6. **Equating the Time**:
The distance traveled in the \(y\)-direction under the influence of the electric field \(E\) can be expressed as:
\[ d_y = vt = \frac{qE}{m} t \]
Setting this equal to the distance traveled in the \(x-z\) plane gives:
\[ nT = \frac{qE}{m} nT \]

**Hint**: Simplify the equation to find a relationship between \(E\), \(B\), and the specific charge \(q/m\).

7. **Final Condition**:
After simplification, we find that the particle will return to the origin if:
\[ \frac{B}{\pi E} = n \]
where \(n\) is an integer.

**Hint**: Check the options provided in the question to see which one matches this condition.

### Conclusion:
The particle will definitely return to the origin once if the condition \(\frac{B}{\pi E}\) is an integer. Therefore, the correct option is \(C\).