Question

The electric potential energy between a proton and an electron is given by $U=U_{0}ln\frac{r}{r_0}$, where $r_0$ is a constant. Assuming Bohr's model to be applicable, write the variation of ${}^{\prime }r{}^{\prime }$ with $n$, $n$ being the principal quantum number.

• A. $r_n\propto n$
• B. $r_n\propto 1/n$
• C. $r_n\propto n^2$
• D. $r_n\propto 1/n^2$

Answer 1

The correct option is A $r_n\propto n$

The electrostatic interaction force between electron and proton is $F=\frac{dU}{dr}=\frac{U_0}{r{r}_0}$

According Bohr's first postulate, $\frac{m{v}^2}{r}=F=\frac{U_0}{r{r}_0}$

or $v^2=\frac{U_0}{m{r}_0}\to constant$

According Bohr's second postulate, $mv{r}_n=\frac{nh}{2\pi }$

As m,v and h are constant so we can write, $r_n\propto n$