An electron of mass `m` and charge `e` is accelerated from rest through a potential difference `V` in vacuum. The final speed of the electron will be

Options

A. `V sqrt((e)/(m))`
B. `sqrt((eV)/(m))`
C. `sqrt((2eV)/(m))`
D. `(2eV)/(m)`

Answer

To find the final speed of an electron that has been accelerated from rest through a potential difference \( V \), we can use the principle of conservation of energy. Here’s a step-by-step solution:

### Step 1: Understand the Energy Conversion
When the electron is accelerated through a potential difference \( V \), the electrical potential energy is converted into kinetic energy. The work done on the electron by the electric field is equal to the change in kinetic energy.

### Step 2: Write the Energy Equations
The potential energy gained by the electron when it is accelerated through a potential difference \( V \) is given by:
\[ \text{Potential Energy} = eV \]
where \( e \) is the charge of the electron.

The kinetic energy (\( KE \)) of the electron when it reaches its final speed \( v \) is given by:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the electron.

### Step 3: Set Up the Conservation of Energy Equation
Since the electron starts from rest, its initial kinetic energy is zero. Therefore, the conservation of energy can be expressed as:
\[ \text{Initial Kinetic Energy} + \text{Potential Energy} = \text{Final Kinetic Energy} \]
This simplifies to:
\[ 0 + eV = \frac{1}{2} mv^2 \]

### Step 4: Rearrange the Equation
Rearranging the equation to solve for \( v \):
\[ eV = \frac{1}{2} mv^2 \]
Multiplying both sides by 2 gives:
\[ 2eV = mv^2 \]

### Step 5: Solve for \( v \)
Now, divide both sides by \( m \):
\[ v^2 = \frac{2eV}{m} \]
Taking the square root of both sides gives:
\[ v = \sqrt{\frac{2eV}{m}} \]

### Final Answer
Thus, the final speed of the electron is:
\[ v = \sqrt{\frac{2eV}{m}} \]

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