The electron gun in an old TV picture tube accelerates electrons between two parallel plates 1.2 cm apart with a 25 kV potential difference between them. The electrons enter through a small hole in the negative plate, accelerate, then exit through a small hole in the positive plate. Assume that the holes are small enough not to affect the electric field or potential. With what speed does an electron exit the electron gun if its entry speed is close to zero?

Verified step by step guidance

1. Identify the given values: The distance between the plates is 1.2 cm (convert to meters: 0.012 m), the potential difference is 25 kV (convert to volts: 25000 V), and the initial speed of the electron is approximately zero.
2. Understand the relationship between the potential difference and the work done on the electron. The work done on the electron by the electric field is equal to the change in its kinetic energy. Use the formula: $qV=\frac{1}{2}m{v}^2$, where $q$ is the charge of the electron, $V$ is the potential difference, $m$ is the mass of the electron, and $v$ is the final speed.
3. Rearrange the formula to solve for the final speed $v$: $v=\sqrt{\frac{2qV}{m}}$. Here, $q=\; -1.6\; \times \; 10-19C$ (magnitude of the electron charge) and $m=\; 9.11\; \times \; 10-31kg$ (mass of the electron).
4. Substitute the known values into the formula: $v=\sqrt{\frac{2(1.6\times 10⁻19)\left(25000\right)}{9.11\times 10⁻31}}$. Perform the calculations step by step to find the final speed.
5. Interpret the result: The final speed of the electron is the value obtained from the above calculation. This speed represents how fast the electron exits the electron gun after being accelerated by the electric field.

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 converts this potential energy into kinetic energy. The relationship between potential difference (V) and energy (E) is given by E = qV, where q is the charge of the electron. This concept is crucial for determining the speed of the electron after acceleration.

Kinetic Energy

Kinetic energy is the energy of an object due to its motion, defined by the equation KE = 1/2 mv², where m is mass and v is velocity. In the context of the electron gun, as the electron accelerates through the electric field, its potential energy is converted into kinetic energy. Understanding this relationship allows us to calculate the final speed of the electron after it exits the plates.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the electric potential energy gained by the electron as it moves through the electric field is transformed into kinetic energy. This principle is fundamental for solving the problem, as it allows us to equate the initial potential energy to the final kinetic energy to find the exit speed of the electron.