The thickness at the centre of a plano convex lens is 3 mm and the diameter is 6 cm. If the speed of light in the material of the lens is $$2 \times 10^8$$ m s$$^{-1}$$. The focal length of the lens is:

For a plano-convex lens, one surface is flat (radius $$\infty$$) and the other is curved with radius $$R$$. The diameter is 6 cm, so the radius of the lens aperture is $$r = 3$$ cm = 0.03 m. The thickness at the centre is $$t = 3$$ mm = 0.003 m.

Using the geometry of the curved surface, for a spherical surface of radius $$R$$ with a chord of half-length $$r$$ and sagitta (depth) $$t$$, we have $$R = \frac{r^2}{2t} + \frac{t}{2}$$. Since $$t \ll r$$, we approximate $$R \approx \frac{r^2}{2t} = \frac{(0.03)^2}{2 \times 0.003} = \frac{9 \times 10^{-4}}{6 \times 10^{-3}} = 0.15$$ m.

The refractive index of the lens material is $$\mu = \frac{c}{v} = \frac{3 \times 10^8}{2 \times 10^8} = 1.5$$. Using the lensmaker's equation for a plano-convex lens: $$\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) = (1.5 - 1)\left(\frac{1}{0.15} - 0\right) = 0.5 \times \frac{1}{0.15} = \frac{0.5}{0.15} = \frac{10}{3}$$.

Therefore $$f = \frac{3}{10} = 0.3$$ m = 30 cm. The correct answer is option 4: 30 cm.