Electron at rest are accelerated by a potential of V volt. These electrons enter the region of space having a uniform, perpendicular magnetic induction field B. the radius of the path of the electrons inside the magnetic field is

Options

A. `(1)/(B) sqrt((mV)/(e ))`
B. `(1)/(B) sqrt((2mV)/(e ))`
C. `(V)/(B)`
D. `(1)/(B) sqrt((v)/(e ))`

Answer

To find the radius of the path of electrons inside a magnetic field after being accelerated by a potential difference \( V \), we can follow these steps:

### Step 1: Calculate the Kinetic Energy of the Electron
When an electron is accelerated through a potential difference \( V \), it gains kinetic energy given by:
\[ KE = eV \]
where \( e \) is the charge of the electron.

### Step 2: Relate Kinetic Energy to Velocity
The kinetic energy can also be expressed in terms of the mass \( m \) of the electron and its velocity \( v \):
\[ KE = \frac{1}{2} mv^2 \]
Setting the two expressions for kinetic energy equal gives:
\[ eV = \frac{1}{2} mv^2 \]

### Step 3: Solve for Velocity
Rearranging the equation to solve for \( v \):
\[ mv^2 = 2eV \implies v^2 = \frac{2eV}{m} \implies v = \sqrt{\frac{2eV}{m}} \]

### Step 4: Determine the Magnetic Force
When the electron enters the magnetic field \( B \), it experiences a magnetic force given by:
\[ F = qvB \]
For an electron, \( q = -e \), so the magnitude of the force is:
\[ F = e v B \]

### Step 5: Centripetal Force and Radius of Circular Motion
The magnetic force acts as the centripetal force that keeps the electron moving in a circular path. This centripetal force is given by:
\[ F = \frac{mv^2}{r} \]
Setting the magnetic force equal to the centripetal force:
\[ e v B = \frac{mv^2}{r} \]

### Step 6: Solve for the Radius \( r \)
Rearranging the equation to solve for the radius \( r \):
\[ r = \frac{mv}{eB} \]

### Step 7: Substitute for \( v \)
Now substitute the expression for \( v \) from Step 3 into the equation for \( r \):
\[ r = \frac{m \sqrt{\frac{2eV}{m}}}{eB} \]
This simplifies to:
\[ r = \frac{\sqrt{2m eV}}{eB} \]

### Final Expression for the Radius
Thus, the radius of the path of the electrons inside the magnetic field is given by:
\[ r = \frac{\sqrt{2m eV}}{eB} \]