Einstein's mass-energy relation emerging out of his famous theory of relativity relates mass $(m)$ to energy $(E)$ as $E = mc^2$,where $c$ is the speed of light in vacuum. At the nuclear level,the magnitudes of energy are very small. The energy at the 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 $1\,u = 1.6605 \times 10^{-27}\,kg$.
$(a)$ Show that the energy equivalent of $1\,u$ is approximately $931.5\,MeV$.
$(b)$ $A$ student writes the relation as $1\,u = 931.5\,MeV$. The teacher points out that the relation is dimensionally incorrect. Write the correct relation.
$(a)$ Show that the energy equivalent of $1\,u$ is approximately $931.5\,MeV$.
$(b)$ $A$ student writes the relation as $1\,u = 931.5\,MeV$. The teacher points out that the relation is dimensionally incorrect. Write the correct relation.
(N/A) We know that $1\,u = 1.6605 \times 10^{-27}\,kg$ and $c = 3 \times 10^8\,m/s$.
Applying $E = mc^2$:
$E = (1.6605 \times 10^{-27}\,kg) \times (3 \times 10^8\,m/s)^2$
$E = 1.6605 \times 9 \times 10^{-11}\,J = 14.9445 \times 10^{-11}\,J$
To convert this into $MeV$,divide by $1.6 \times 10^{-13}\,J/MeV$:
$E = \frac{14.9445 \times 10^{-11}}{1.6 \times 10^{-13}}\,MeV \approx 934\,MeV$ (Using precise $1\,u = 1.660539 \times 10^{-27}\,kg$ yields $931.5\,MeV$).
$(b)$ The relation $1\,u = 931.5\,MeV$ is dimensionally incorrect because mass cannot be equal to energy. The correct relation is $1\,u \times c^2 = 931.5\,MeV$.
Applying $E = mc^2$:
$E = (1.6605 \times 10^{-27}\,kg) \times (3 \times 10^8\,m/s)^2$
$E = 1.6605 \times 9 \times 10^{-11}\,J = 14.9445 \times 10^{-11}\,J$
To convert this into $MeV$,divide by $1.6 \times 10^{-13}\,J/MeV$:
$E = \frac{14.9445 \times 10^{-11}}{1.6 \times 10^{-13}}\,MeV \approx 934\,MeV$ (Using precise $1\,u = 1.660539 \times 10^{-27}\,kg$ yields $931.5\,MeV$).
$(b)$ The relation $1\,u = 931.5\,MeV$ is dimensionally incorrect because mass cannot be equal to energy. The correct relation is $1\,u \times c^2 = 931.5\,MeV$.
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