The figure shows a transparent sphere of radius $R$ and refractive index $\mu$. An object $O$ is placed at a distance $x$ from the pole of the first surface so that a real image is formed at the pole of the exactly opposite surface. If $x = 2R$,then the value of $\mu$ is

What is the focal length of this curved surface in $cm$?

As shown in the figure,region $BCDEF$ and $ABFG$ have refractive indices $2.0$ and $1.5$ respectively. $A$ particle $O$ is kept at the midpoint of region $BCEF$. The final image of the object as seen by the eye is at a distance of:

Two identical glass rods $S_1$ and $S_2$ (refractive index $= 1.5$) have one convex end of radius of curvature $10 \ cm$. They are placed with the curved surfaces at a distance $d$ as shown in the figure,with their axes (shown by the dashed line) aligned. When a point source of light $P$ is placed inside rod $S_1$ on its axis at a distance of $50 \ cm$ from the curved face,the light rays emanating from it are found to be parallel to the axis inside $S_2$. The distance $d$ is (in $cm$)

$A$ concave spherical refracting surface separates two media,glass and air $(\mu_1 = 1.5, \mu_2 = 1.0)$. If the image is to be real,at what minimum distance $u$ should the object be placed in the glass if $R$ is the radius of curvature?

Light from a point source in air falls on a spherical glass surface ($n = 1.5$ and radius of curvature $R = 20 \ cm$). The distance of the light source from the glass surface is $100 \ cm$. Find the image distance. (in $cm$)