$A$ plano-convex lens of refractive index ${\mu _1}$ and focal length $f_1$ is kept in contact with another plano-concave lens of refractive index ${\mu _2}$ and focal length $f_2$. If the radius of curvature of their spherical faces is $R$ each and $f_1 = 2f_2$,then ${\mu _1}$ and ${\mu _2}$ are related as:

Two lenses having focal lengths in the ratio $f_1:f_2 = 2:3$ are combined to produce no net dispersion. Find the ratio of the dispersive powers of the glasses used.

$A$ convex lens of focal length $20 \, cm$ is cut into two equal parts. This results in two plano-convex lenses as shown in the figure. These two parts are then placed in contact with each other as shown in the figure. What will be the focal length of the system in $cm$?

The focal length (in $cm$) of a lens with dispersive power $0.45$ that should be placed in contact with a convex lens of focal length $84 \,cm$ and dispersive power $0.21$ to form an achromatic combination is:

$(a)$ Determine the 'effective focal length' of the combination of two lenses,a convex lens of focal length $30 \; cm$ and a concave lens of focal length $20 \; cm$,if they are placed $8.0 \; cm$ apart with their principal axes coincident. Does the answer depend on which side of the combination a beam of parallel light is incident? Is the notion of effective focal length of this system useful at all?
$(b)$ An object $1.5 \; cm$ in size is placed on the side of the convex lens in the arrangement $(a)$ above. The distance between the object and the convex lens is $40 \; cm$. Determine the magnification produced by the two-lens system,and the size of the image.

The plane faces of two identical plano-convex lenses,each with focal length $f$,are pressed against each other using an optical glue to form a convex lens. The distance from the optical centre at which an object must be placed to obtain an image of the same size as the object is