Consider the process of pair production. |
It is possible for a photon to materialize into an electron and a positron. In this way, electromagnetic energy is converted into matter. Both energy and linear momentum are conserved when an electron-positron pair is created near an atomic nucleus. In absence of nucleus, pair production cannot occur in empty space because |
A) it is impossible to conserve momentum without presence of nucleus
B) it is impossible to conserve energy without presence of nucleus
C) Both a and b are correct
D) pair production is possible even in empty space
Correct Answer: C
Solution :
Following is the vector diagram of the momenta involved if a photon were to materialise into an electron-positron pair in empty space. |
For momentum to conserve, |
\[\frac{hf}{c}=2p\cos \theta \] |
\[\Rightarrow \] \[hf=2pc\cdot \cos \theta \] |
(here, f= frequency of incident photon) |
As, \[p=mv\] |
\[hf=2m{{c}^{2}}\left( \frac{v}{c} \right)\cos \theta \] |
As \[\frac{v}{c}<1\]land \[\cos \theta \le 1\] |
We can say that, | ||
\[hf<2m{{c}^{2}}\] | ... (i) | |
But energy conservation required | ||
\[hf=2m{{c}^{2}}\] | ... (ii) | |
Hence, it is impossible to satisfy both | ||
Eqs. (i) and (ii) simultaneously. | ||
But if there is some other object is present to carry away part of initial momentum, then both condition (i) and (ii) can be satisfied. | ||
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