Optical Instruments

Optical instruments use combinations of lenses and mirrors to extend the capabilities of the human eye. The two most important compound optical instruments are the microscope and the telescope.

The Compound Microscope

A compound microscope uses two converging lenses:

• Objective lens: short focal length $f_{\text{obj}}$, creates a magnified real image of the specimen
• Eyepiece lens: acts as a magnifying glass, enlarges the intermediate image

The total magnification is:

$M_{\text{microscope}} = \frac{L}{f_{\text{obj}}} \times \frac{D}{f_{\text{eye}}}$

Where:

• $L$ is the tube length — the distance between the objective's back focal point and the eyepiece's front focal point (typically 160 mm)
• $D$ is the near-point distance — the closest comfortable viewing distance for the human eye (typically 250 mm)
• $f_{\text{obj}}$ and $f_{\text{eye}}$ are the focal lengths of the objective and eyepiece

Example: Objective $f_{\text{obj}} = 4\,\text{mm}$, eyepiece $f_{\text{eye}} = 25\,\text{mm}$, $L = 160\,\text{mm}$, $D = 250\,\text{mm}$:

$M = \frac{160}{4} \times \frac{250}{25} = 40 \times 10 = 400$

The Refracting Telescope

A refracting telescope also uses two lenses, but designed for distant objects:

$M_{\text{telescope}} = -\frac{f_{\text{obj}}}{f_{\text{eye}}}$

The negative sign indicates an inverted image (astronomical telescopes give an inverted view). The angular magnification magnitude is:

$|M| = \frac{f_{\text{obj}}}{f_{\text{eye}}}$

A large objective focal length and short eyepiece focal length gives high magnification. Telescopes also gather more light with larger objective diameters.

Your Task

Implement the magnification formulas for both instruments. For the microscope, use a default near-point distance of 250 mm.