A particle of charge `q` and mass `m` moving with a velocity `v` along the x-axis enters the region `xgt0` with uniform magnetic field `B` along the `hatk` direction. The particle will penetrate in this region in the `x`-direction upto a distance `d` equal to

Options

A. Zero
B. `(mv)/(qB)`
C. `(2mv)/(qB)`
D. Infinity

Answer

To solve the problem of a charged particle moving in a magnetic field, we can follow these steps:

### Step 1: Understand the Motion of the Particle
The particle of charge \( q \) and mass \( m \) is moving with a velocity \( v \) along the x-axis. When it enters the region where \( x > 0 \), it encounters a uniform magnetic field \( \vec{B} \) directed along the \( \hat{k} \) direction (which is the z-direction). The magnetic field will exert a force on the charged particle, causing it to move in a circular path.

**Hint:** Remember that the magnetic force acts perpendicular to the velocity of the charged particle, causing circular motion.

### Step 2: Identify the Force Acting on the Particle
The magnetic force \( \vec{F} \) acting on the particle can be given by the Lorentz force equation:
\[ \vec{F} = q \vec{v} \times \vec{B} \]
Since the velocity \( \vec{v} \) is along the x-axis and the magnetic field \( \vec{B} \) is along the z-axis, the force will act in the y-direction.

**Hint:** Use the right-hand rule to determine the direction of the magnetic force.

### Step 3: Determine the Radius of the Circular Path
The particle will move in a circular path due to the magnetic force. The radius \( R \) of this circular path can be derived from the balance of the magnetic force and the centripetal force required for circular motion. The centripetal force is given by:
\[ F_c = \frac{mv^2}{R} \]
Setting the magnetic force equal to the centripetal force:
\[ qvB = \frac{mv^2}{R} \]
From this equation, we can solve for the radius \( R \):
\[ R = \frac{mv}{qB} \]

**Hint:** Make sure to isolate \( R \) correctly by rearranging the equation.

### Step 4: Relate the Distance \( d \) to the Radius
In this scenario, the distance \( d \) that the particle penetrates into the region while moving in the x-direction is equal to the radius \( R \) of the circular path.

Thus, we can conclude:
\[ d = R = \frac{mv}{qB} \]

**Hint:** Remember that the distance traveled in the x-direction corresponds to the radius of the circular path due to the magnetic force.

### Final Answer
The distance \( d \) that the particle will penetrate in the x-direction is given by:
\[ d = \frac{mv}{qB} \]