A) \[{{\lambda }_{\text{0}}}\text{t}\]
B) \[{{\lambda }_{0}}\left( 1+\frac{e{{E}_{0}}}{m{{V}_{0}}}t \right)\]
C) \[\frac{{{\lambda }_{0}}}{\left( 1+\frac{e{{E}_{0}}}{m{{V}_{0}}}t \right)}\]
D) \[{{\lambda }_{0}}\]
Correct Answer: C
Solution :
[c] Initial de-Broglie wavelength |
\[{{\lambda }_{0}}\text{=}\frac{\text{h}}{\text{m}{{\text{V}}_{\text{0}}}}\] (i) |
Acceleration of electron |
\[a=\frac{e{{E}_{0}}}{m}\] |
Velocity after time 't' |
\[\text{V=}\left( {{\text{V}}_{\text{0}}}\text{+}\frac{\text{e}{{\text{E}}_{\text{0}}}}{\text{m}}\text{t} \right)\] |
So, \[\lambda \text{=}\frac{\text{h}}{\text{mV}}\text{=}\frac{\text{h}}{\text{m}\left( {{\text{V}}_{\text{0}}}\text{+}\frac{\text{e}{{\text{E}}_{\text{0}}}}{\text{m}}\text{t} \right)}\] |
\[=\frac{h}{m{{V}_{0}}\left[ 1+\frac{e{{E}_{0}}}{m{{V}_{0}}}t \right]}=\frac{{{\lambda }_{0}}}{\left[ 1+\frac{e{{E}_{0}}}{m{{V}_{0}}}t \right]}\] (ii) |
Divide (ii) by (i), |
\[\lambda =\frac{{{\lambda }_{0}}}{\left[ 1+\frac{e{{E}_{0}}}{m{{V}_{0}}}t \right]}\] |
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