Question

If a photon and an electron have the same De-Broglie wavelength, then?
A.Both have same kinetic energy
B.Proton has more K.E. than electron
C.Electron has more K.E. than proton
D.Both have same velocity

Answer 1

Hint: Start by writing the De-Broglie wavelength equation ,also relation between kinetic energy and linear momentum and deduce relation between wavelength and energy. Compare the relation formed using masses of proton and electron, finally representing it in terms of energy.

Complete answer:
We know, De-Broglie wavelength is given by the relation
$\lambda = \dfrac{{\text{h}}}{{\text{p}}} \to (1)$, where ‘h’ is the Planck’s constant and ‘p’ is the linear momentum.
Also, we know the relation between Linear momentum and kinetic energy which is
$K.E. = \dfrac{{{p^2}}}{{2m}}$, where K.E. is kinetic energy and ‘m’ is the mass.
If we rearrange the above equation we get,
$p = \sqrt {2m(K.E.)} $
So , relation 1 becomes,
$\lambda = \dfrac{{\text{h}}}{{\sqrt {2m(K.E.)} }} \to eqn.2$
And it is given that a photon and an electron have the same De-Broglie wavelength .
Let De-Broglie wavelength of photon and electron be ${\lambda _p}$and ${\lambda _e}$ , their masses be ${m_p}$ and ${m_e}$, and their kinetic energies be $K.E{._p}$ and $K.E{._e}$
 $\therefore {\lambda _p} = {\lambda _e}$
Using equation 2 , we get
$
  \dfrac{{\text{h}}}{{\sqrt {2{m_p}{{(K.E.)}_p}} }} = \dfrac{{\text{h}}}{{\sqrt {2{m_e}{{(K.E.)}_e}} }} \\
   \Rightarrow {m_p}{(K.E.)_p} = {m_e}{(K.E.)_e} \\
   \Rightarrow \dfrac{{{m_p}}}{{{m_e}}} = \dfrac{{{{(K.E.)}_e}}}{{{{(K.E.)}_p}}} \\
$
We know \[{m_p} > {m_e} \Rightarrow \dfrac{{{m_p}}}{{{m_e}}} > 1\]
$
   \Rightarrow \dfrac{{{{(K.E.)}_e}}}{{{{(K.E.)}_p}}} > 1 \\
  \therefore {(K.E.)_e} > {(K.E.)_p} \\
$
Which means the Kinetic Energy of the electron is higher than that of proton for the same De-Broglie wavelength.

So, Option C is the correct answer.

Note:
Similar questions can be asked for the same kinetic energy or linear momentum , In that case use the same relations as above and deduce the required relation from the same. Students must remember the values of charge of electron , proton , neutron and their masses as well, as they are very much required for the comparison .