The potential difference required to accelerate an electron from rest to a given velocity depends directly on the electron’s change in kinetic energy. The kinetic energy of an electron is given by:

## Kinetic Energy of an Electron

KE = 1/2 x m x v^{2}

Where:

- KE is the kinetic energy in Joules (J)
- m is the mass of the electron = 9.11 x 10
^{-31}kg - v is the velocity of the electron in m/s

An electron at rest has no kinetic energy. To accelerate it to a velocity v, we must impart kinetic energy equal to the above equation. This kinetic energy is given by the potential difference according to:

## Kinetic Energy from Electric Potential

KE = e x V

Where:

- KE is the kinetic energy in Joules
- e is the charge of an electron = -1.60 x 10
^{-19}C - V is the potential difference in Volts

By equating these two expressions, we can derive the potential difference required to accelerate an electron to a given velocity:

## Potential Difference to Accelerate Electron

V = (1/2 x m x v^{2}) / e

Plugging in the known values for electron mass and charge:

V = (1/2 x 9.11 x 10^{-31} x v^{2}) / -1.60 x 10^{-19}

V = 5.69 x 10^{12} x v^{2} Volts

This equation gives the potential difference in volts required to accelerate an electron from rest to a velocity v in m/s. For example, to accelerate an electron to 10% the speed of light (3 x 10^{7} m/s):

V = 5.69 x 10^{12} x (3 x 10^{7})^{2} = 5.10 x 10^{3} Volts

So a potential difference of 5.10 kilovolts would give an electron a velocity of 30 million m/s.

## Numerical Examples

Here are some example velocity values and the required potential differences:

Velocity v (m/s) | Potential Difference V (Volts) |
---|---|

1 x 10^{6} |
5.69 x 10^{3} |

1 x 10^{7} |
5.69 x 10^{6} |

3 x 10^{7} |
5.10 x 10^{3} |

1 x 10^{8} |
5.69 x 10^{9} |

This table shows that very large potential differences, on the order of megavolts, are required to accelerate electrons to appreciable velocities. This is why particle accelerators like the Large Hadron Collider require enormous electromagnets and power supplies to accelerate particles to relativistic velocities.

## Non-Relativistic vs Relativistic Velocity

It’s important to note that the above equations apply only for non-relativistic velocities, where v 8 m/s), relativistic effects become important. The kinetic energy and required potential difference are modified by the relativistic gamma factor:

Gamma (γ) = 1 / √(1 – v^{2}/c^{2})

Thus for relativistic electron acceleration, the kinetic energy and required potential difference become:

KE = γ x (1/2)mv^{2}

V = γ x (1/2)mv^{2}/e

Where the gamma factor γ increases dramatically as the velocity approaches c. This means accelerating electrons to near light-speed requires enormous potential differences on the scale of billions or trillions of volts.

## Conclusion

In summary, the potential difference required to accelerate an electron from rest to a given velocity v can be calculated using the equations:

V = (1/2 x m x v^{2}) / e

V = 5.69 x 10^{12} x v^{2} Volts (for an electron)

This shows that accelerating electrons to fast but non-relativistic velocities requires potential differences on the order of thousands to millions of volts. Approaching the speed of light c requires enormous potential differences due to relativistic effects. Particle accelerators use sophisticated high-voltage equipment, giant magnets, radiofrequency cavities, and miles-long tunnels to achieve these high energies.

### References

- Griffiths, D. J. (1999). Introduction to Electrodynamics (3rd ed.). Prentice Hall.
- Jackson, J. D. (1998). Classical Electrodynamics (3rd ed.). John Wiley & Sons.
- Young, H. D., & Freedman, R. A. (2016). University Physics with Modern Physics (14th ed.). Pearson.