Question

An electron orbiting around the nucleus of an atom

• A. Has a magnetic dipole moment
• B. Exerts an electric force on the nucleus equal to that on it by the nucleus
• C. Does produce a magnetic induction at the nucleus
• D. Has a net energy, whose magnitude is inversely proportional to its distance from the nucleus.

Answer 1

Answer: (A), (B), (C), (D)

Has a net energy, whose magnitude is inversely proportional to its distance from the nucleus.

Given
$r\to$ radius of atom
$v\to$ velocity of electron

Magnetic moment
$M=IA$
Using
$\left\{\begin{array}{cc}T=\frac{2\pi r}{v}& A=\pi r^2\\ I=\frac{q}{t}& \because I=\frac{ev}{2\pi r}\end{array}\right\}$

$M=\frac{ev}{2\pi r}\times \pi r^2=\frac{evr}{2}$

From Bohr’s first postulates
$\frac{m{v}^2}{r}=\frac{kZ{e}^2}{r^2}$
$\Rightarrow \frac{m{v}^2}{2}=\frac{kZ{e}^2}{2r}$
Now mechanical energy
$M.E.=\frac{m{v}^2}{2}-\frac{kZ{e}^2}{r}$
$M.E.=\frac{kZ{e}^2}{2r}-\frac{kZ{e}^2}{r}=\frac{-kZ{e}^2}{2r}$
$\therefore$ Net mechanical energy $\propto \frac{1}{r}$