On the EM Spectrum, which type of wave has the most energy?

A general guideline tends to be : short wavelength, high energy . But here is a way to show which waves are the most energetic:

The energy of a wave is given by the equation:

#E=hf#

#h#=Planck's constant #(6,6261·10^(-34) Js^-1)#
#f#=frequency of the wave

Hence we can see that the energy of a wave is proportional to its frequency, as the other term is a constant.

Then we can ask ourselves, which waves are the ones with the highest frequency?

If we use another equation:

#c=flambda#

#c#=speed of light ,#3.0 times 10^8 ms^-1#
#f#=frequency (Hz)
#lambda#=wavelength in meters.

Then we can see that, as #c# is constant in a vacuum, and #f# is high, then #lambda#, the wavelength, must be low.
Now if we use this diagram of the EM-spectrum which show wavelengths:

We can thus conclude that the waves that have the shortest wavelength are gamma rays, and thus they are the most energetic because they must also have the highest frequency.