Question
In a hypothetical system, a particle of mass $$m$$ and charge $$-3q$$ is moving around a very heavy particle charge $$q.$$ Assume that Bohr's model is applicable to this system, then velocity of mass $$m$$ in the first orbit is
A.
$$\frac{{3{q^2}}}{{2{\varepsilon _0}h}}$$
B.
$$\frac{{3{q^2}}}{{4{\varepsilon _0}h}}$$
C.
$$\frac{{3q}}{{2\pi {\varepsilon _0}h}}$$
D.
$$\frac{{3q}}{{4\pi {\varepsilon _0}h}}$$
Answer :
$$\frac{{3{q^2}}}{{2{\varepsilon _0}h}}$$
Solution :
$$\eqalign{
& \frac{{m{v^2}}}{r} = \frac{{3{q^2}}}{{4\pi {\varepsilon _0}{r^2}}} \cr
& \Rightarrow mvr = \frac{{3{q^2}}}{{4\pi {\varepsilon _0}v}}\,......\left( {\text{i}} \right) \cr
& {\text{and}}\,\,\frac{{nh}}{{2\pi }} = mvr\,......\left( {{\text{ii}}} \right) \cr} $$
Using (i) and (ii) putting $$n=1$$
$$\eqalign{
& \frac{h}{{2\pi }} = \frac{{3{q^2}}}{{4\pi {\varepsilon _0}v}} \cr
& \Rightarrow v = \frac{{3{q^2}}}{{2{\varepsilon _0}h}} \cr} $$