Relativistic corrections become necessary when the expression for the kinetic energy `1/2mv^(2)`, becomes comparable with `mc^(2)`, where m is the mass of the particle. At what de-broglie wavelength will relativistic corrections become important for an electron?

Options

A. `lamda=10nm`
B. `lamda=10^(-1)nm`
C. `lamda=10^(-4)nm`
D. `lamda=10^(-6)nm`

Explanation

de-Brolige wavelength
`" "lamda=h/(mv)impliesv=h/(mlamda)`
Here, `" "h=6.6xx10^(-34)Js`
and for electron, `" "m=9xx10^(-31)kg`
Now consider each option one by one
(a) `" "lamda_(1)=10nm=10xx10^(-9)m=10^(-8)m`
`implies" "v_(1)=(6.6xx10^(-34))/((9xx10^(131))xx10^(-8))`
`implies" "=(2.2)/(3)xx10^(5)~~10^(5)m//s`
(b) `lamda_(2)=10^(-1)nm=10^(-1)xx10^(-9)m=10^(-10)m`
`implies" "v_(2)=(6.6xx10^(-34))/((9xx10^(-31))xx10^(-10))=10^(7)m//s`
(c) `lamda_(3)=10^(-4)nm=10^(-4)xx10^(-9)m=10^(-13)m`
`implies" "v_(3)=(6.6xx10^(-34))/((9xx10^(-31))xx10^(-13))~~10^(10)m//s`
(d) `lamda_(4)=10^(-6)nm=10^(-6)xx10^(-9)m=10^(-9)m=10^(-15)m`
`implies" "v_(4)=(6.6xx10^(-34))/(9xx10^(-31)xx10^(-15))~~10^(12)m//s`
Thus options (c) and (d) are correct as `v_(3) and v_(4)` is greater than `3xx10^(8)m//s.`