Optical Instruments

The Eye as a Starting Point

The human eye is a natural optical instrument. It uses a lens to form a real, inverted image on the retina. Optical instruments like microscopes and telescopes are designed to extend the eye's capability — to see very small objects (microscope) or very distant objects (telescope). Both instruments use combinations of lenses to produce a magnified, virtual image that the eye can view comfortably.

The Simple Microscope (Magnifying Glass)

A simple microscope is just a single convex lens of short focal length ($f$). The object is placed between the lens and its focal point ($u < f$). The lens then produces a virtual, erect, and magnified image on the same side as the object.

Angular Magnification ($M$)

The magnifying power is defined as the ratio of the angle subtended by the image at the eye ($\beta$) to the angle subtended by the object when placed at the near point (the closest distance for clear vision, $D = 25\ \text{cm}$) without the instrument ($\alpha$).

Case 1: Image at the near point ($v = -D$)

The eye is most strained but gets maximum magnification. Using the lens formula $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ with $v = -D$ and $u = -u$ (real object), we get:

The angular magnification is:

• $D$ = least distance of distinct vision ($25\ \text{cm}$)
• $f$ = focal length of the lens (in cm)
• This is the maximum magnifying power.

Case 2: Image at infinity ($v = \infty$)

The eye is relaxed. The object is placed exactly at the focal point ($u = f$). The angular magnification becomes:

• This is the normal adjustment magnifying power, smaller than the previous case.

The Compound Microscope

A compound microscope uses two convex lenses: an objective (short focal length, small aperture) and an eyepiece (short focal length, larger aperture). The objective produces a real, inverted, and magnified intermediate image of the object. The eyepiece then acts as a simple microscope to magnify this intermediate image.

Magnification

The total angular magnification ($M$) is the product of the lateral magnification by the objective ($m_o$) and the angular magnification by the eyepiece ($m_e$):

Objective magnification:

• $v_o$ = image distance from objective
• $u_o$ = object distance from objective
• $h'$ = height of intermediate image
• $h$ = height of object

Eyepiece magnification (image at infinity, relaxed eye):

• $f_e$ = focal length of eyepiece
• $D = 25\ \text{cm}$

Total magnification (normal adjustment):

For maximum magnification (image at near point of eye):

So:

Key points:

• The objective forms a real, inverted image inside the tube.
• The eyepiece magnifies this image.
• The final image is virtual and inverted with respect to the object.
• The length of the microscope tube ($L$) is approximately $v_o + f_e$ (when image is at infinity).

The Astronomical Telescope (Refracting)

A telescope is used to view distant objects. It consists of an objective (large focal length, large aperture) and an eyepiece (short focal length). The objective forms a real, inverted image of the distant object at its focal plane. The eyepiece magnifies this image.

Angular Magnification ($M$)

For a distant object, the angle subtended at the objective is $\alpha \approx \frac{h}{f_o}$ (where $h$ is the height of the image formed by the objective). The eyepiece produces a virtual image at infinity, so the angle subtended at the eye is $\beta \approx \frac{h}{f_e}$.

$M = \frac{\beta}{\alpha} = \frac{f_o}{f_e}$ …