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 \(\dfrac{\alpha}{10}\). The value of \(\alpha\) is _______.

Sunlight comes from infinity and is focused at the principal focus of the convex lens. The paper burns in the shortest time when the intensity of light is maximum, which occurs at the focal point.

Therefore, the focal length of the convex lens is \(f = 30\) cm.

Assuming the lens is equiconvex, the radii of curvature are \(R_1 = 60\) cm and \(R_2 = -60\) cm.

Using the Lens Maker's formula:

\(\dfrac{1}{f} = (\mu - 1) \left( \dfrac{1}{R_1} - \dfrac{1}{R_2} \right)\)

Substituting the given values:

\(\dfrac{1}{30} = (\mu - 1) \left( \dfrac{1}{60} - \left( -\dfrac{1}{60} \right) \right)\)

\(\dfrac{1}{30} = (\mu - 1) \left( \dfrac{2}{60} \right)\)

\(\dfrac{1}{30} = (\mu - 1) \left( \dfrac{1}{30} \right)\)

\(\mu - 1 = 1 \Rightarrow \mu = 2\)

Given that the refractive index is \(\dfrac{\alpha}{10}\):

\(\dfrac{\alpha}{10} = 2 \Rightarrow \alpha = 20\)

Answer: \(20\)