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Question
In a gamma decay process, the internal energy of a nucleus of mass M decreases, a gamma photon of energy E and linear momentum E/c is emitted and the nucleus recoils. Find the decrease in internal energy.
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Solution
Let the nucleus recoils with a velocity v.
Applying the law of conservation of linear momentum, we get:
Linear momentum of recoiled nucleus = Linear momentum of gamma photon
⇒ mv = \[\frac{E}{c}\]
∴ \[v = \frac{E}{mc}\]
Kinetic energy of the recoiled nucleus = \[\frac{1}{2}M v^2\]
\[\Rightarrow K . E . = \frac{1}{2}m \left( \frac{E}{mc} \right)^2 = \frac{1}{2}\frac{E^2}{m c^2}\]
Decrease in the internal energy = photon energy + the kinetic energy of the recoiled nucleus
⇒ Decrease in the internal energy = \[E + \frac{E^2}{2m c^2}\]
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