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. `(eV)/(2m)`
B. `(eV)/(m)`
C. `sqrt((2eV)/(m)`
D. `sqrt((ev)/(2m))`

Answer

To find the final speed of an electron 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 an electron is accelerated through a potential difference \( V \), the electrical potential energy gained by the electron is converted into kinetic energy.

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

### Step 3: Write the Expression for Kinetic Energy
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 4: Apply Conservation of Energy
According to the conservation of energy, the potential energy gained by the electron will equal the kinetic energy it possesses at its final speed:
\[ eV = \frac{1}{2} mv^2 \]

### Step 5: Rearrange the Equation to Solve for \( v \)
To find the final speed \( v \), we can rearrange the equation:
\[ \frac{1}{2} mv^2 = eV \]
Multiplying both sides by 2 gives:
\[ mv^2 = 2eV \]
Now, divide both sides by \( m \):
\[ v^2 = \frac{2eV}{m} \]
Taking the square root of both sides, we find:
\[ v = \sqrt{\frac{2eV}{m}} \]

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

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