A charged particle with charge to mass ratio `((q)/(m)) = (10)^(3)/(19) Ckg^(-1)` enters a uniform magnetic field `vec(B) = 20 hat(i) + 30 hat(j) + 50 hat(k) T` at time t = 0 with velocity `vec(V) = (20 hat(i) + 50 hat(j) + 30 hat(k)) m//s`. Assume that magnetic field exists in large space.
The pitch of the helical path of the motion of the particle will be

Options

A. `pi//100m`
B. `pi//125m`
C. `pi//215m`
D. `pi//250m`

Explanation

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

### Step 1: Understand the Components of Motion
The motion of a charged particle in a magnetic field is helical. The pitch of the helix is the distance between two consecutive turns of the helix. The motion can be broken down into two components:
- Circular motion due to the magnetic field.
- Linear motion along the direction of the magnetic field.

### Step 2: Define Pitch
The pitch \( P \) of the helical path can be expressed as:
\[ P = V \cos \theta \cdot T \]
where \( V \) is the velocity of the particle, \( \theta \) is the angle between the velocity vector and the magnetic field vector, and \( T \) is the time period of the circular motion.

### Step 3: Calculate the Time Period \( T \)
The time period \( T \) for the circular motion of a charged particle in a magnetic field is given by:
\[ T = \frac{2\pi m}{qB} \]
where \( m \) is the mass of the particle, \( q \) is the charge of the particle, and \( B \) is the magnitude of the magnetic field.

### Step 4: Calculate the Dot Product \( B \cdot V \)
The dot product \( B \cdot V \) gives us the component of the velocity in the direction of the magnetic field. Given:
\[ \vec{B} = 20 \hat{i} + 30 \hat{j} + 50 \hat{k} \quad \text{(T)} \]
\[ \vec{V} = 20 \hat{i} + 50 \hat{j} + 30 \hat{k} \quad \text{(m/s)} \]
We can calculate \( B \cdot V \):
\[ B \cdot V = (20)(20) + (30)(50) + (50)(30) = 400 + 1500 + 1500 = 3400 \, \text{(T m/s)} \]

### Step 5: Calculate the Magnitude of the Magnetic Field \( B \)
The magnitude of the magnetic field \( B \) is given by:
\[ B = \sqrt{20^2 + 30^2 + 50^2} = \sqrt{400 + 900 + 2500} = \sqrt{3800} \approx 61.64 \, \text{(T)} \]

### Step 6: Substitute Values into the Pitch Formula
Now, substituting \( B \cdot V \) and \( B \) into the pitch formula:
\[ P = \frac{B \cdot V}{B} \cdot \frac{2\pi m}{qB^2} \]
We know that the charge-to-mass ratio \( \frac{q}{m} = \frac{10^3}{19} \, \text{C kg}^{-1} \), thus:
\[ \frac{m}{q} = \frac{19}{10^3} \]
Substituting this into the pitch formula:
\[ P = \frac{3400}{61.64} \cdot \frac{2\pi \cdot \frac{19}{10^3}}{(61.64)^2} \]

### Step 7: Calculate the Final Value
Calculating the above expression gives:
\[ P \approx 0.10681 \, \text{m} \approx 0.107 \, \text{m} \]

### Final Answer
The pitch of the helical path of the motion of the particle is approximately:
\[ \boxed{0.107 \, \text{m}} \]