(II) What is the speed of an electron after being accelerated from rest by 28,000 V?

Verified step by step guidance

1. Identify the relationship between the work done on the electron by the electric field and the kinetic energy gained by the electron. The work done is equal to the charge of the electron multiplied by the potential difference: \( W = q \Delta V \).
2. Recognize that the work done on the electron is converted entirely into its kinetic energy, since it starts from rest. Therefore, \( W = KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the electron and \( v \) is its final speed.
3. Substitute \( W = q \Delta V \) into the kinetic energy equation: \( q \Delta V = \frac{1}{2}mv^2 \).
4. Rearrange the equation to solve for the speed \( v \): \( v = \sqrt{\frac{2q \Delta V}{m}} \).
5. Substitute the known values: \( q = 1.6 \times 10^{-19} \; \text{C} \) (charge of the electron), \( \Delta V = 28,000 \; \text{V} \), and \( m = 9.11 \times 10^{-31} \; \text{kg} \) (mass of the electron). Simplify the expression to find \( v \).

Key Concepts

Electric Potential Energy

Electric potential energy is the energy a charged particle possesses due to its position in an electric field. When an electron is accelerated through a potential difference, it gains kinetic energy equal to the work done on it by the electric field, which can be calculated using the formula: KE = qV, where q is the charge of the electron and V is the voltage.

Kinetic Energy

Kinetic energy is the energy of an object due to its motion, defined mathematically as KE = 1/2 mv², where m is the mass and v is the velocity of the object. In the context of the question, the kinetic energy gained by the electron after being accelerated can be equated to the electric potential energy it acquired from the voltage applied.

Charge of an Electron

The charge of an electron is a fundamental physical constant, approximately -1.6 x 10^-19 coulombs. This negative charge is crucial for calculations involving electric fields and potential differences, as it determines the direction of the force acting on the electron when it is subjected to an electric field, influencing its acceleration and final speed.