question_answer 75)

                  Einstein?s mass ? energy relation emerging out of his famous theory of relativity relates mass \[(m)\] to energy \[(E)\] as \[E=m{{c}^{2}},\] where c is speed of light in vacuum. At the nuclear lever, the magnitudes of energy are very small. The energy at nuclear level is usually measured in MeV, where 1 MeV \[=1.6\,\times \,{{10}^{-13}}J;\] the masses are measured in unified atomic mass unit \[(u)\] where \[1u=1.67\times {{10}^{-27}}\,kg\]. (a) Show that the energy equivalent of 1 u is 931.5 MeV. (b) A student writes the relation as 1 u is 931.5 MeV. The teacher points out that the relation is dimensionally incorrect.                 Write the correct relation.

Answer:

                  (a) \[E=\,m{{c}^{2}},\] where \[m=1\,u\]                 \[=\,1.67\,\times \,{{10}^{-27}}\,kg\]                 and \[c\,=\,3\,\times \,{{10}^{8}}\,m{{s}^{-1}}\]                 \[\therefore \] \[E\,=\,1.67\,\times \,{{10}^{-27\,}}\times \,\,9\,\times \,{{10}^{16}}\]                 \[=15.03\,\times \,{{10}^{-11}}J\]                 Since \[1\,\,MeV\,=\,1.6\,\times \,{{10}^{-13}}J\]                 \[\therefore \]\[E\,=\,\frac{15.03\,\times \,{{10}^{-11}}\,J}{1.6\,\times \,{{10}^{-13\,}}}=\,931.5\,MeV\]                                (b) \[1\,u\,=\,931.5\,MeV\]                 L.H.S represents mass and R.H.S represents energy                 Since \[\,m\,=\,\frac{E}{{{c}^{2}}}\]                 \[\therefore \]\[1\,u\,=\,931.5\,MeV/{{c}^{2}}\]