A charged particle of specific charge `alpha` moves with a velocity `vecv=v_0hati` in a magnetic field `vecB=(B_0)/(sqrt2)(hatj+hatk)`. Then (specific charge=charge per unit mass)

Options

A. path of the particle is a heix
B. path of the particle is a circle
C. distance moved by the particle in time `t=(pi)/(B_(0)alpha)` is `(pi)/(B_(0))`
D. velocity of particle after time `t=(pi)/(B_(0)alpha)` is `(v_(0)/(2)hati+(v_(0))/(2))`

Answer

To solve the problem, we need to analyze the motion of a charged particle in a magnetic field. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Given Information
We have a charged particle with a specific charge `α`, moving with a velocity vector:
\[ \vec{v} = v_0 \hat{i} \]
and it is placed in a magnetic field given by:
\[ \vec{B} = \frac{B_0}{\sqrt{2}} (\hat{j} + \hat{k}). \]

### Step 2: Determine the Direction of Motion
The velocity vector is in the x-direction, while the magnetic field has components in the y and z directions. This means that the velocity vector and the magnetic field vector are perpendicular to each other.

### Step 3: Analyze the Motion of the Charged Particle
When a charged particle moves in a magnetic field and the velocity is perpendicular to the magnetic field, it experiences a magnetic force that acts as a centripetal force. This results in circular motion.

### Step 4: Calculate the Magnetic Force
The magnetic force \(\vec{F}\) on a charged particle is given by:
\[ \vec{F} = q (\vec{v} \times \vec{B}), \]
where \(q\) is the charge of the particle. The specific charge \(\alpha\) is defined as:
\[ \alpha = \frac{q}{m}, \]
where \(m\) is the mass of the particle.

### Step 5: Determine the Radius of Circular Motion
The magnetic force provides the centripetal force required for circular motion:
\[ F = \frac{mv^2}{r}, \]
where \(r\) is the radius of the circular path. Setting the magnetic force equal to the centripetal force gives:
\[ q v B = \frac{mv^2}{r}. \]
Rearranging this equation allows us to express the radius \(r\) in terms of the specific charge \(\alpha\):
\[ r = \frac{mv}{qB} = \frac{v}{\alpha B}. \]

### Step 6: Calculate the Distance Traveled
The distance traveled by the particle in one complete circular motion can be calculated using the circumference of the circle:
\[ \text{Distance} = 2\pi r = 2\pi \left(\frac{v}{\alpha B}\right). \]

### Step 7: Substitute the Values
Substituting the values of \(v\) and \(B\):
\[ \text{Distance} = 2\pi \left(\frac{v_0}{\alpha \cdot \frac{B_0}{\sqrt{2}}}\right) = \frac{2\pi \sqrt{2} v_0}{\alpha B_0}. \]

### Conclusion
Thus, we have established that the path of the particle is circular and calculated the distance traveled in one complete revolution. The specific charge is crucial in determining the radius of the circular path.