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प्रश्न
Derive an expression for de Broglie wavelength of electrons.
Obtain an expression for the de-Broglie wavelength associated with an electron accelerated from rest through a potential difference of ‘V’ volts.
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उत्तर
An electron of mass m is accelerated through a potential difference of V volts. The kinetic energy acquired by the electron is given by:
`1/2 mv^2` = eV
Therefore, the speed (v) of the electron is:
v = `sqrt((2 eV)/m)`
Hence, the de-Broglie wavelength of the electron is:
λ = `h/(mv)`
= `h/sqrt (2 e m V)`
Substituting the known values in the above equation, we get,
λ = `(6.626 xx 10^-34)/(sqrt (2 V xx 1.6 xx 10^-19 xx 9.11 xx 10^-31))`
= `(12.27 xx 10^-10)/sqrt V` meter
= `12.27/sqrt V` Å
For example, if an electron is accelerated through a potential difference of 100 V, then its de-Broglie wavelength is 1.227 Å.
Since the kinetic energy of the electron, K = eV, then the de-Broglie wavelength associated with the electron can also be written as:
λ = `h/(sqrt (2 m K))`
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