The electronvolt and work done on electrons

​​In a nutshell

An electronvolt is the energy gained by an electron accelerated from rest through a potential difference of $$1\ V$$ in a vacuum; it has a value of $$1.6\times10^{-19} \ J$$. When a charged particle is accelerated by a potential difference, the work done on the particle turns into kinetic energy.

Equations

Constants

Variable definitions

The electronvolt

When measuring energies the joule ends up being too large for particles hence a smaller energy unit was created, the electronvolt. It is derived from the equation:

​$$W=VQ$$​​

When an electron travels through a potential difference, its energy is equal to $$VQ$$. If the potential difference is $$1V$$ and since the electron has a charge of $$1.6\times10^{-19}\ C$$, the work done is  $$1V\times e =1.6\times10^{-19} \ J$$. This is the energy of one electronvolt; it is defined as the energy gained by one electron accelerated from rest through a potential difference of $$1\ V$$ in a vacuum.

​$$1\ eV= 1.6\times10^{-19}\ J$$​​

Example

An electron is accelerated through a potential difference from rest and gains $$3\ keV$$ of energy. What is its energy in joules?

Firstly, write down what you know:

​$$E=3\ keV$$​​

Just like for kilometers or kilowatts, the $$k$$ stands for $$\times10^{3}$$, hence you can write:

$$E=3\ keV=3000\ eV$$​ 

Next, write down the conversion of electronvolts to joules:

$$1\ eV=1.6\times10^{-19}\ J$$​​

The energy in joules will therefore be:

$$E= 3000\times1.6\times10^{-19}$$​​

Calculating the value:

$$E=4.8\times10^{-16}\ J$$​​

The energy gained by the electron is $$\underline{4.8\times10^{-16}\ J}$$.

Kinetic energy of charges

When charged particles are accelerated through a potential difference the work done on them turns into kinetic energy:

​$$\begin {aligned}W&=KE \\VQ&=\dfrac{1}{2}mv^2\end{aligned}$$​​

For electrons this equation can be rewritten using the elementary charge $$e$$:

$$eV=\dfrac{1}{2}m_ev^2$$​​

Using this you can for example find the speed of electrons in a beam.

Example

An electron is accelerated through a potential difference of $$10\ V$$, what is the final speed of the electron?

Firstly, write down what you know:

$$V=10\ V$$​​

Next, write down the equation:

$$eV=\dfrac{1}{2}m_ev^2$$​​

Rearrange to find $$v$$:

$$v=\sqrt{\dfrac{2eV}{m_e}}$$​​

Substitute all the values into the equation:

$$v=\sqrt{\dfrac{2\times1.6\times10^{-19}\times10}{9.11\times10^{-31}}}$$​​

Calculate the speed:

$$v=1.9\times10^{6}\ m\ s^{-1}$$​​

The final speed of the electron is $$\underline{1.9\times10^{6} \ m\ s^{-1}}$$.