In an electron microscope, a beam of electrons is produced by a device called an electron gun. In the electron gun, electrons are 'boiled off' from a heated filament (which is the cathode and held at high negative voltage) and are attracted towards an anode. The high voltage difference Δ*V *causes the electrons to accelerate.

(i) Assuming that the speed of the electrons is zero as they are emitted from the filament and that as they reach the anode it is *v, *find an expression relating the final speed *v *to the charge on an electron, the mass of an electron and Δ*V*, and hence calculate the speed *v *of the electrons after passing through a voltage difference of 120 kV. You should identify clearly the three main steps in your answer ('Decide how you are going to tackle the problem'; 'Do the calculation'; 'Check that your answer makes sense') and you should state any assumptions that you make. You should show all of your working.

(ii) The wavelength, *λ*, of a beam of electrons is related to the speed, *v*, and mass, *m*, of the electrons by the following equation:

*λ*, = h/mv

* *where *h *is the Planck constant*. *Use this equation and your answer to part (i) to show that the wavelength λ of the electrons is related to the high voltage difference Δ*V *by the following equation:

^{}

*λ*^{2}, = h^{2}/2emΔ*V*

(iii) Calculate the value of the wavelength of a beam of electrons in a microscope operated at 120 kV as in (i) and also at 15 kV. Your working should show explicitly how you arrived at units of length (metres) for your answers.

(iv) Compare the speed you calculated in part (i) with the speed of light and hence comment on the reasons why your calculated results may be inaccurate.