In an electron gun, the electrons are accelerated by the potential `V`. If the `e` is the charge and `m` is the mass of the electron, then the maximum velocity of these electrons will be

Options

A. (a)`(2eV)/m`
B. (b)`sqrt((2eV)/m)`
C. (c)`sqrt((2m)/(eV))`
D. (d)`V^2/(2 em)`

Answer

To find the maximum velocity of electrons in an electron gun accelerated by a potential \( V \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Energy Relation**:
The kinetic energy gained by the electrons when they are accelerated through a potential \( V \) is given by:
\[ \text{Kinetic Energy} = eV \]
where \( e \) is the charge of the electron.

2. **Kinetic Energy Formula**:
The kinetic energy of an object can also be expressed in terms of its mass \( m \) and velocity \( v \):
\[ \text{Kinetic Energy} = \frac{1}{2} mv^2 \]

3. **Setting the Two Expressions Equal**:
Since both expressions represent the kinetic energy of the electrons, we can set them equal to each other:
\[ eV = \frac{1}{2} mv^2 \]

4. **Solving for Velocity**:
To find the maximum velocity \( v \), we rearrange the equation:
\[ mv^2 = 2eV \]
Now, divide both sides by \( m \):
\[ v^2 = \frac{2eV}{m} \]
Taking the square root of both sides gives us:
\[ v = \sqrt{\frac{2eV}{m}} \]

5. **Final Expression**:
Thus, the maximum velocity \( v \) of the electrons is:
\[ v = \sqrt{\frac{2eV}{m}} \]