A particle of specific charge `q/m = pi C kg^-1` is projected from the origin toward positive x-axis with a velocity of `10 ms^-1` in a uniform magnetic field `vec B = -2 hat k T.` The velocity `vec v` of particle after time `t=1/12 s` will be (in `ms^-1`)

Options

A. `5[hati+sqrt3hatj]`
B. `5[sqrt3hati+hatj]`
C. `5[sqrt3hati-hatj]`
D. `5[hati+hatj]`

Explanation

To find the velocity of the particle after time \( t = \frac{1}{12} \) seconds, we will follow these steps:

### Step 1: Understand the given parameters
- Specific charge \( \frac{q}{m} = \pi \, \text{C/kg} \)
- Initial velocity \( \vec{v} = 10 \, \text{m/s} \) along the positive x-axis
- Magnetic field \( \vec{B} = -2 \hat{k} \, \text{T} \)
- Time \( t = \frac{1}{12} \, \text{s} \)

### Step 2: Determine the angular velocity
The angular velocity \( \omega \) of the particle in a magnetic field is given by:
\[ \omega = \frac{qB}{m} \]
Substituting the specific charge \( \frac{q}{m} \):
\[ \omega = \frac{q}{m} B = \pi \cdot (-2) = -2\pi \, \text{rad/s} \]

### Step 3: Calculate the angle turned in time \( t \)
The angle \( \theta \) turned by the particle in time \( t \) is given by:
\[ \theta = \omega t \]
Substituting the values:
\[ \theta = -2\pi \cdot \frac{1}{12} = -\frac{\pi}{6} \, \text{radians} \]

### Step 4: Find the components of the velocity
Since the particle moves in a circular path, we can find the new components of the velocity after time \( t \).

- The new velocity components can be expressed as:
\[ V_x = V \cos(\theta) \]
\[ V_y = V \sin(\theta) \]

Substituting \( V = 10 \, \text{m/s} \) and \( \theta = -\frac{\pi}{6} \):
\[ V_x = 10 \cos\left(-\frac{\pi}{6}\right) = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3} \, \text{m/s} \]
\[ V_y = 10 \sin\left(-\frac{\pi}{6}\right) = 10 \cdot \left(-\frac{1}{2}\right) = -5 \, \text{m/s} \]

### Step 5: Write the final velocity vector
The velocity vector after time \( t \) is given by:
\[ \vec{V} = V_x \hat{i} + V_y \hat{j} = 5\sqrt{3} \hat{i} - 5 \hat{j} \, \text{m/s} \]

### Final Answer
Thus, the velocity of the particle after time \( t = \frac{1}{12} \, \text{s} \) is:
\[ \vec{V} = 5\sqrt{3} \hat{i} - 5 \hat{j} \, \text{m/s} \]

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