Question 16: Atoms & Nuclei

Question

A proton has kinetic energy E = 100 eV which is equal to that of a photon. The wavelength of photon is $$\lambda$$2 and that of proton is $$\lambda$$1. The ratio $${{{\lambda _2}} \over {{\lambda _1}}}$$ is proportional to

Accepted Answer

Kinetic energy of a proton is given by
$$E = {1 \over 2}{m_p}v_p^2$$
$$ \Rightarrow E = {{{p^2}} \over {2{m_p}}}$$ [$$\because$$ $$p = mv$$]
$$	herefore$$ Wavelength of proton is, $${\lambda _1} = {h \over p}$$
$$ \Rightarrow {\lambda _1} = {h \over {\sqrt {2{m_p}E} }}$$ ..... (i)
Now, energy of photon is given by
$$E = h{v_2} \Rightarrow E = {{hc} \over {{\lambda _2}}}$$
$$ \Rightarrow {\lambda _2} = {{hc} \over E}$$ ....... (ii)
where, $$\lambda$$
2
is wavelength of photon.
On dividing Eq. (ii) by Eq. (i), we get
$${{{\lambda _2}} \over {{\lambda _1}}} = {{hc} \over E} 	imes {{\sqrt {2{m_p}E} } \over h}$$
$$ \Rightarrow {{{\lambda _2}} \over {{\lambda _1}}} = {c \over {\sqrt E }}\sqrt {2{m_p}} $$
$$ \Rightarrow {{{\lambda _2}} \over {{\lambda _1}}} \propto {E^{ - 1/2}}$$

Answer

E2

Answer

E$$^{{1 \over 2}}$$

Answer

E$$-$$1

Answer

E$$^{ - {1 \over 2}}$$

Atoms & Nuclei

A proton has kinetic energy E = 100 eV which is equal to that of a photon. The wavelength of photon is $\lambda2 and that of proton is \lambda1. The ratio {{{\lambda _2}} \over {{\lambda _1}}}$ is proportional to

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