Question

The speed of an electron accelerated from rest under a potential difference \( V \) is

• A. proportional to \( V \)
• B. proportional to \( \sqrt{V} \)
• C. proportional to \( \frac{1}{V} \)
• D. proportional to \( V^2 \)

Hint:

For an electron accelerated through a potential difference, its speed is proportional to the square root of the potential difference.

Answer 1

The Correct Option is B

Step 1: Energy of the electron.
The kinetic energy gained by the electron when accelerated through a potential difference \( V \) is given by: \[ K.E. = eV \] where \( e \) is the charge of the electron.
Step 2: Relationship with speed.
The kinetic energy of the electron is also given by: \[ K.E. = \frac{1}{2}mv^2 \] where \( m \) is the mass of the electron and \( v \) is its velocity.
Step 3: Equating the two expressions.
Equating the kinetic energy expressions, we get: \[ eV = \frac{1}{2}mv^2 \] Solving for \( v \), we get: \[ v = \sqrt{\frac{2eV}{m}} \] Thus, the speed is proportional to \( \sqrt{V} \).
Step 4: Elimination.
- (A) proportional to \( V \): Incorrect, the speed is not directly proportional to \( V \).
- (B) proportional to \( \sqrt{V} \): Correct, as derived above.
- (C) proportional to \( \frac{1}{V} \): Incorrect.
- (D) proportional to \( V^2 \): Incorrect.
Step 5: Conclusion.
Hence, the speed of the electron is proportional to \( \sqrt{V} \).