Cross-section view of a prism is the equilateral triangle $$ABC$$ shown in the figure. The minimum deviation is observed using this prism when the angle of incidence is equal to the prism angle. The time taken by light to travel from P (midpoint of BC) to A is _________ $$\times 10^{-10}$$ s. (Given, speed of light in vacuum = $$3 \times 10^8$$ m s$$^{-1}$$ and $$\cos 30° = \frac{\sqrt{3}}{2}$$)

We need to calculate the time taken by light to travel along the altitude $$PA$$ inside an equilateral triangular prism under the given minimum deviation conditions

1. Find the Refractive Index of the Prism ($$\mu$$)

For an equilateral triangle prism, the prism angle is:

$$A = 60^\circ$$

We are given that at minimum deviation, the angle of incidence ($$i$$) is equal to the prism angle ($$A$$):

$$i = A = 60^\circ$$

In the condition of minimum deviation, the angle of refraction ($$r$$) inside the prism relates to the prism angle by:

$$r = \frac{A}{2} = \frac{60^\circ}{2} = 30^\circ$$

Applying Snell's Law at the first refracting surface:

$$\mu = \frac{\sin(i)}{\sin(r)} = \frac{\sin(60^\circ)}{\sin(30^\circ)} = \frac{\left(\frac{\sqrt{3}}{2}\right)}{\left(\frac{1}{2}\right)} = \sqrt{3}$$

2. Calculate the Speed of Light Inside the Prism ($$v$$)

The speed of light inside a medium depends on its refractive index:

$$v = \frac{c}{\mu} = \frac{3 \times 10^8}{\sqrt{3}} = \sqrt{3} \times 10^8\text{ m/s}$$

3. Determine the Distance Travelled ($$PA$$)

From the given image layout, the side length of the equilateral triangle cross-section is $$10\text{ cm}$$. The point $$P$$ is the midpoint of base $$BC$$, making $$PA$$ the altitude of $$\triangle ABC$$:

$$PA = \text{Side} \times \cos(30^\circ) = 10\text{ cm} \times \frac{\sqrt{3}}{2} = 5\sqrt{3}\text{ cm} = 5\sqrt{3} \times 10^{-2}\text{ m}$$

4. Calculate the Travel Time ($$t$$)

Using the uniform motion formula ($$\text{time} = \frac{\text{distance}}{\text{speed}}$$):

$$t = \frac{PA}{v} = \frac{5\sqrt{3} \times 10^{-2}\text{ m}}{\sqrt{3} \times 10^8\text{ m/s}}$$

$$\text{Cancelling out } \sqrt{3}:$$

$$t = 5 \times 10^{-2} \times 10^{-8} = 5 \times 10^{-10}\text{ s}$$

Final Whole Number Answer: $$5$$