The focal lengths of the objective and the eyepiece of a compound microscope are $2 \,cm$ and $3 \,cm$ respectively, and the distance between them is $15 \,cm$. The final image formed by the eyepiece is at infinity. The distances of the object and the image produced by the objective lens from the objective lens are respectively:

To improve the resolving power of a compound microscope, we should:

An object viewed from a near point distance of $25 \, cm$,using a microscopic lens with magnification $6$,gives an unresolved image. $A$ resolved image is observed at infinite distance with a total magnification double the earlier,using an eyepiece along with the given lens and a tube of length $0.6 \, m$. The focal length of the eyepiece is equal to $.... \, cm$.

In a microscope of tube length $10 \ cm$, two convex lenses are arranged with focal lengths of $2 \ cm$ and $5 \ cm$. The total magnification obtained with this system for normal adjustment is $(5)^{k}$. The value of $k$ is . . . . . . .

$A$ compound microscope has a magnifying power of $30$. The focal length of its eyepiece is $5 \ cm$. Assuming the final image is at the least distance of distinct vision $(25 \ cm)$,the magnification produced by the objective is:

Answer the following questions:
$(a)$ The angle subtended at the eye by an object is equal to the angle subtended at the eye by the virtual image produced by a magnifying glass. In what sense then does a magnifying glass provide angular magnification?
$(b)$ In viewing through a magnifying glass,one usually positions one's eyes very close to the lens. Does angular magnification change if the eye is moved back?
$(c)$ Magnifying power of a simple microscope is inversely proportional to the focal length of the lens. What then stops us from using a convex lens of smaller and smaller focal length and achieving greater and greater magnifying power?
$(d)$ Why must both the objective and the eyepiece of a compound microscope have short focal lengths?
$(e)$ When viewing through a compound microscope,our eyes should be positioned not on the eyepiece but a short distance away from it for best viewing. Why? How much should be that short distance between the eye and eyepiece?