The focal length of a convex lens is $30 \ cm$ and the size of the image is a quarter of the object,then the object distance is ........ $cm$.

An object approaches a convergent lens from the left of the lens with a uniform speed $5 \, m/s$ and stops at the focus. The image:

$A$ thin lens of refractive index $1.5$ has an optical power of $-5 \ D$ in air. Its optical power in a liquid medium with a refractive index of $1.6$ is $D$.

$A$ light ray hits the pole of a thin biconvex lens as shown in the figure. The angle made by the emergent ray with the optic axis will be

Explain how the image is formed by a thin convex lens and derive the lens maker's formula: $\frac{1}{f} = (n_{21} - 1) \left[ \frac{1}{R_1} - \frac{1}{R_2} \right]$.

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