The ratio of speeds of electromagnetic waves in vacuum and a medium, having dielectric constant $$k = 3$$ and permeability of $$ \mu = 2 \mu_{0}$$, is
($$\mu_{0}$$ = permeability of vacuum)

We need to determine the ratio of the speed of electromagnetic waves in vacuum to that in a medium whose dielectric constant is $$K = 3$$ and permeability is $$\mu = 2\mu_0$$.

First, recall that the speed of an electromagnetic wave in a medium is given by $$v = \frac{1}{\sqrt{\mu \epsilon}}$$, while in vacuum it is $$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$$. When the medium has relative permeability $$\mu_r$$ and dielectric constant $$K$$, we have $$\mu = \mu_r \mu_0$$ and $$\epsilon = K \epsilon_0$$.

Next, forming the ratio of these speeds yields $$\frac{c}{v} = \frac{\sqrt{\mu \epsilon}}{\sqrt{\mu_0 \epsilon_0}} = \sqrt{\frac{\mu \epsilon}{\mu_0 \epsilon_0}} = \sqrt{\mu_r \, K}$$. Since $$\mu = 2 \mu_0$$, the relative permeability is $$\mu_r = 2$$.

Substituting the given values gives $$\frac{c}{v} = \sqrt{2 \times 3} = \sqrt{6}$$, so that the speeds are in the ratio $$c : v = \sqrt{6} : 1$$.

The correct answer is Option (2): $$\sqrt{6} : 1$$.