If the radius of the $$3^{rd}$$ Bohr's orbit of hydrogen atom is $$r_3$$ and the radius of $$4^{th}$$ Bohr's orbit is $$r_4$$. Then

We need to find the relationship between the radius of the 3rd and 4th Bohr orbits of a hydrogen atom.

The radius of the $$n^{th}$$ orbit in a hydrogen atom is given by $$r_n = a_0 \cdot n^2$$, where $$a_0$$ is the Bohr radius. Thus the radius is proportional to $$n^2$$.

Substituting $$n=3$$ yields $$r_3 = a_0 \cdot (3)^2 = 9a_0$$, and substituting $$n=4$$ yields $$r_4 = a_0 \cdot (4)^2 = 16a_0$$.

Taking the ratio, we find $$\frac{r_4}{r_3} = \frac{16a_0}{9a_0} = \frac{16}{9}$$, which implies $$r_4 = \frac{16}{9} r_3$$.

Therefore, the correct answer is Option B: $$r_4 = \frac{16}{9}r_3$$.