A positive charge q is projected in magnetic field of width `(mv)/(sqrt(2)qB)` with velocity v . Then, the time taken by charged particle to emerge from the magnetic field is

Options

A. `(m)/(sqrt(2)qB)`
B. `(pim)/(4qB)`
C. `(pim)/(2qB)`
D. `(npi)/(sqrt(2)*qB)`

Explanation

To solve the problem of finding the time taken by a charged particle to emerge from a magnetic field, we can follow these steps:

### Step 1: Understand the Situation
A positive charge \( q \) is projected into a magnetic field with a width \( d = \frac{mv}{\sqrt{2}qB} \) and with a velocity \( v \). The magnetic field is perpendicular to the velocity of the charge.

### Step 2: Determine the Radius of Circular Motion
When a charged particle moves in a magnetic field, it experiences a magnetic force that acts as a centripetal force, causing it to move in a circular path. The radius \( r \) of this circular motion can be expressed as:
\[ r = \frac{mv}{qB} \]

### Step 3: Compare the Radius and Width
Given that the width of the magnetic field \( d \) is less than the radius \( r \) (since \( d = \frac{mv}{\sqrt{2}qB} \)), we can conclude that the particle will not complete a full circular path but will instead trace out a part of a circle.

### Step 4: Calculate the Angle Subtended
The angle \( \theta \) subtended at the center of the circular path when the particle exits the magnetic field can be determined using the sine function:
\[ \sin(\theta) = \frac{d}{r} \]
Substituting for \( d \) and \( r \):
\[ \sin(\theta) = \frac{\frac{mv}{\sqrt{2}qB}}{\frac{mv}{qB}} = \frac{1}{\sqrt{2}} \]
Thus, we find:
\[ \theta = \frac{\pi}{4} \]

### Step 5: Calculate the Time Taken
The time period \( T \) for a complete circular motion is given by:
\[ T = \frac{2\pi m}{qB} \]
Since the particle only travels through an angle of \( \frac{\pi}{4} \), the time \( t \) taken to travel through this angle is:
\[ t = \frac{\theta}{2\pi} \cdot T = \frac{\frac{\pi}{4}}{2\pi} \cdot \frac{2\pi m}{qB} \]
Simplifying this gives:
\[ t = \frac{m}{qB} \cdot \frac{1}{4} \]

### Final Answer
Thus, the time taken by the charged particle to emerge from the magnetic field is:
\[ t = \frac{\pi m}{4qB} \]