A student wants to make a very small particle accelerator using a 9.0 V battery. What speed will an electron have after being accelerated from rest through the 9.0 V potential difference?

Verified step by step guidance

1. Understand the relationship between the potential difference and the kinetic energy gained by the electron. The work done on the electron by the electric field is equal to the change in its kinetic energy. This can be expressed as: $qV\; =\; \backslash (\backslash frac\{1\}\{2\}\backslash )mv^2$, 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 of the electron.
2. Rearrange the equation to solve for the speed $v$: $v\; =\; \backslash (\backslash sqrt\{\backslash frac\{2qV\}\{m\}\backslash )\}$.
3. Substitute the known values into the equation. The charge of the electron is $q\; =\; 1.6\; \backslash (\backslash times\backslash )\; 10^\{-19\}\; \backslash (\backslash text\{\; C\}\backslash )$, the potential difference is $V\; =\; 9.0\; \backslash (\backslash text\{\; V\}\backslash )$, and the mass of the electron is $m\; =\; 9.11\; \backslash (\backslash times\backslash )\; 10^\{-31\}\; \backslash (\backslash text\{\; kg\}\backslash )$.
4. Perform the calculation inside the square root: calculate $2qV$ and divide it by $m$.
5. Take the square root of the result to find the final speed $v$ of the electron.

Key Concepts

Electric Potential Energy

Electric potential energy is the energy a charged particle possesses due to its position in an electric field. When a charged particle, like an electron, moves through a potential difference, it gains energy equal to the product of its charge and the voltage. This energy is converted into kinetic energy as the particle accelerates.

Kinetic Energy

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

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 voltage is converted entirely into kinetic energy, enabling us to relate the voltage to the speed of the electron.