A charged particle (q.m) released from origin with velocity `v=v_(0)hati` in a uniform magnetic field `B=(B_(0))/(2)hati+(sqrt3B_(0))/(2)hatJ`.
Pitch of the helical path described by the particle is

Options

A. `(2pimv_(0))/(B_(0)q)`
B. `(sqrt3pimv_(0))/(2B_(0)q)`
C. `(pimv_(0))/(B_(0)q)`
D. `(2sqrt3pimv_(0))/(B_(0)q)`

Answer

To find the pitch of the helical path described by a charged particle in a uniform magnetic field, we can follow these steps:

### Step 1: Understand the Given Information
We have a charged particle with:
- Charge \( q \)
- Mass \( m \)
- Initial velocity \( \mathbf{v} = v_0 \hat{i} \)
- Magnetic field \( \mathbf{B} = \frac{B_0}{2} \hat{i} + \frac{\sqrt{3}B_0}{2} \hat{j} \)

### Step 2: Determine the Angle Between Velocity and Magnetic Field
To find the pitch of the helical path, we first need to determine the angle \( \theta \) between the velocity vector \( \mathbf{v} \) and the magnetic field vector \( \mathbf{B} \).

- The components of the magnetic field are:
- \( B_x = \frac{B_0}{2} \)
- \( B_y = \frac{\sqrt{3}B_0}{2} \)

Using the tangent function to find the angle:
\[ \tan \theta = \frac{B_y}{B_x} = \frac{\frac{\sqrt{3}B_0}{2}}{\frac{B_0}{2}} = \sqrt{3} \]
This implies:
\[ \theta = 60^\circ \]

### Step 3: Calculate the Pitch of the Helical Path
The pitch \( P \) of the helical path can be given by the formula:
\[ P = V \cos \theta \times T \]
where \( T \) is the time period of the motion.

#### Step 3.1: Calculate \( V \cos \theta \)
From the velocity:
\[ V = v_0 \]
Using \( \cos 60^\circ = \frac{1}{2} \):
\[ V \cos \theta = v_0 \cos 60^\circ = v_0 \times \frac{1}{2} = \frac{v_0}{2} \]

#### Step 3.2: Calculate the Time Period \( T \)
The time period \( T \) for a charged particle in a magnetic field is given by:
\[ T = \frac{2\pi m}{qB} \]
To find \( B \), we calculate the magnitude of \( \mathbf{B} \):
\[ B = \sqrt{B_x^2 + B_y^2} = \sqrt{\left(\frac{B_0}{2}\right)^2 + \left(\frac{\sqrt{3}B_0}{2}\right)^2} = \sqrt{\frac{B_0^2}{4} + \frac{3B_0^2}{4}} = \sqrt{B_0^2} = B_0 \]
Thus, the time period becomes:
\[ T = \frac{2\pi m}{qB_0} \]

### Step 4: Substitute Values into the Pitch Formula
Now substituting \( V \cos \theta \) and \( T \) into the pitch formula:
\[ P = \left(\frac{v_0}{2}\right) \left(\frac{2\pi m}{qB_0}\right) \]
This simplifies to:
\[ P = \frac{v_0 \pi m}{qB_0} \]

### Final Answer
The pitch of the helical path described by the particle is:
\[ P = \frac{\pi m v_0}{q B_0} \]