Question

In one model of the electron, the electron of mass $m_e$ is thought to be a uniformly charged shell of radius $R$ and to tal charge $e$, whose electrostatic energy $E$ is equivalent to its mass $me via$ Einstein’s mass energy relation $E =m_{e}c^{2}$. In this model, $R$ is approximately $(m_{e} = 9.1 \times 10^{31} \,kg, c = 3 \times108\, ms^{-1}, 1 / 4\pi\varepsilon_{0} = 9 \times 10^{9} \,Fm^{-1},$ magnitude of the electron charge $= 1.6 \times 10^{19} C)$

• A. $14 \times10^{-15} \,m $
• B. $2 \times10^{-13} \,m $
• C. $5.3 \times10^{-11} \,m $
• D. $2.8\times10^{-35} \,m $

Answer 1

Answer: (A)

Electrostatic energy of charged shell is
$E=\frac{e^{2}}{8\pi\varepsilon_{0}R}$
and from Einstein’s mass - energyequivalence, $E = mc^{2}$
Equating both, we get
$\frac{e^{2}}{8\pi\varepsilon_{0}R}=mc^{2}$
$\Rightarrow Rv=\frac{e^{2}}{8\pi\varepsilon_{0 }.mc^{2}}$
Substituting values, we have
$\Rightarrow R= \frac{\left(1.6 \times10^{-19}\right)^{2}\times9\times10^{9}}{2\times9.1\times10^{-31}\times9\times10^{16}}$
$=1.4\times10^{-15} \,m$