Question 215: Ray Optics

Question

If sunlight is focused on a paper using convex lens, it starts burning the paper in shortest time when the lens is kept at 30 cm above the paper. If the radius of curvature of the lens is 60 cm then the refractive index of the lens material is $\frac{\alpha}{10}$. The value of $\alpha$ is ________.

Accepted Answer

For the paper to burn in the shortest time, the image of the Sun must be formed exactly on the paper.
Since the Sun is very far away, the rays coming from it are nearly parallel. A convex lens brings parallel rays to its principal focus. So, the distance of the paper from the lens is equal to the focal length.
Thus,
$$ f = 30 	ext{ cm} $$
Now use the lens makerÃ¢ÂÂs formula for a thin convex lens in air:
$$ \frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}ight) $$
For a symmetric double convex lens,
$$ R_1 = +60 	ext{ cm}, \qquad R_2 = -60 	ext{ cm} $$
So,
$$ \frac{1}{30} = (\mu - 1)\left(\frac{1}{60} - \left(-\frac{1}{60}ight)ight) $$
$$ \frac{1}{30} = (\mu - 1)\left(\frac{2}{60}ight) $$
$$ \frac{1}{30} = (\mu - 1)\left(\frac{1}{30}ight) $$
Hence,
$$ \mu - 1 = 1 $$
$$ \mu = 2 $$
Given that refractive index is $\frac{\alpha}{10}$,
$$ \frac{\alpha}{10} = 2 $$
$$ \alpha = 20 $$
So, the value of $\alpha$ is
$$ \boxed{20} $$

Ray Optics

If sunlight is focused on a paper using convex lens, it starts burning the paper in shortest time when the lens is kept at 30 cm above the paper. If the radius of curvature of the lens is 60 cm then the refractive index of the lens material is \frac{\alpha}{10}. The value of \alpha is ________.

Integer / Numeric Answer Type