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Relativistic corrections become necessar...

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?

A

`lamda=10 nm`

B

`lamda=10^(-1)nm`

C

`lamda=10^(-4)nm`

D

`lamda=10^(-6)nm`

Text Solution

Verified by Experts

The correct Answer is:
C, D
The de Broglie wavelength is given by
`lamda=h/(mv)`
`impliesv=h/(mlamda)`
For, `lamda_1=10 nm = 10xx10^(-9) m = 10 ^(-8)m`
`v_1=(6.63xx10^(-34))/(9.1xx10^(-31)xx10^(-8))`
`=0.728xx10^(5)m//s~=10^5m//s`
For, `lamda_2=10^(-1) nm = 10^(-1)xx10^(-9) m = 10 ^(-10)m`
`v_2=(6.63xx10^(-34))/(9.1xx10^(-31)xx10^(-10))`
`=0.728xx10^(7)m//s~=10^7m//s`
For `lamda_3=10^(-4) nm = 10^(-4)xx10^(-9) m = 10 ^(-13)m`
`v_3=(6.63xx10^(-34))/(9.1xx10^(-31)xx10^(-13))`
`=0.73xx10^(10)m//s~=10^(10)m//s`
For `lamda_4=10^(-6) nm = 10^(-6)xx10^(-9) m = 10 ^(-15)m`
`v_4=(6.63xx10^(-34))/(9.1xx10^(-31)xx10^(-15))`
`=0.73xx10^(12)m//s~=10^(12)m//s`
Velocities `v_3 and v_4` are greater than velocity of light `(=3xx10^8m//s)` . So , relativistic correction is needed for `lamda=10^(-4) nm and 10^(-6) nm.`
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