Two protons are projected simultaneously from a fixed point with the same velocity v into a region, magnetic there exists a uniform magnetic field. The magnetic field strength at B and it si perpendicular to the initial direction of v. One proton starts at time t=0 and another proton at `t = (pim)/(2qB)`. The separation between them at time `t = (pim)/(qB)` (where, m and q are the mass and charge of proton), will be approximately

Options

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

Answer

To solve the problem, we need to analyze the motion of two protons in a magnetic field and calculate their separation at a given time. Here’s a step-by-step breakdown of the solution:

### Step 1: Understanding the Motion in a Magnetic Field
When a charged particle, such as a proton, moves in a magnetic field perpendicular to its velocity, it experiences a Lorentz force that causes it to move in a circular path. The force acting on the proton is given by:
\[ F = qvB \]
where \( q \) is the charge of the proton, \( v \) is its velocity, and \( B \) is the magnetic field strength.

### Step 2: Finding the Radius of Circular Motion
The centripetal force required for circular motion is provided by the magnetic force. Therefore, we can set the magnetic force equal to the centripetal force:
\[ qvB = \frac{mv^2}{r} \]
Rearranging this equation gives us the radius \( r \) of the circular path:
\[ r = \frac{mv}{qB} \]

### Step 3: Determining the Time Period of Revolution
The time period \( T \) for one complete revolution of the proton can be calculated using the formula for the circumference of a circle divided by the velocity:
\[ T = \frac{2\pi r}{v} \]
Substituting the expression for \( r \):
\[ T = \frac{2\pi \left(\frac{mv}{qB}\right)}{v} = \frac{2\pi m}{qB} \]

### Step 4: Analyzing the Timing of the Protons
The first proton starts at \( t = 0 \), and the second proton starts at \( t = \frac{\pi m}{2qB} \). This means that the second proton starts its motion a quarter of the time period after the first proton.

### Step 5: Position of the Protons at Time \( t = \frac{\pi m}{qB} \)
At \( t = \frac{\pi m}{qB} \), which is half the time period \( T \):
- The first proton completes half a revolution and is at the opposite side of the circle.
- The second proton, which started \( \frac{\pi m}{2qB} \) later, completes a quarter revolution.

### Step 6: Calculating the Separation
At \( t = \frac{\pi m}{qB} \):
- The first proton is at the point directly opposite its starting point.
- The second proton is at a point that is \( 90^\circ \) from its starting point.

The separation between the two protons can be visualized as the hypotenuse of a right triangle where both protons are at the ends of the two perpendicular sides (each of length \( r \)):
\[ \text{Separation} = \sqrt{(r)^2 + (r)^2} = \sqrt{2r^2} = r\sqrt{2} \]

### Step 7: Substituting for \( r \)
Substituting \( r = \frac{mv}{qB} \):
\[ \text{Separation} = \sqrt{2} \left(\frac{mv}{qB}\right) = \frac{\sqrt{2} mv}{qB} \]

### Final Answer
The separation between the two protons at time \( t = \frac{\pi m}{qB} \) is approximately:
\[ \frac{\sqrt{2} mv}{qB} \]