lecture 18

Compound Microscope and Telescope

Compound Microscope

The compound microscope uses two lenses: the objective and the eyepiece. The objective forms a real, magnified image of the object, which then acts as the object for the eyepiece. The eyepiece further magnifies this image, allowing for high overall magnification.

• Objective lens (fo): Forms a real, inverted image close to its focal point.

• Eyepiece lens (fe): Acts as a magnifying glass for the image formed by the objective.

• Tube length (L): Distance between the objective and eyepiece focal points.

• Near point (N): Closest distance at which the eye can focus comfortably (typically 25 cm).

Angular magnification (M) of a compound microscope is given by:

$M \approx -\frac{L}{f_o} \cdot \frac{N}{f_e}$

where $L$ is the tube length, $f_o$ is the focal length of the objective, $N$ is the near point, and $f_e$ is the focal length of the eyepiece.

• Example: For $f_o = 0.40$ cm, $f_e = 3.0$ cm, $L = 20.0$ cm, $N = 25$ cm:

• $M \approx -\frac{20.0}{0.40} \cdot \frac{25}{3.0} = -420$

The maximum magnification using the eyepiece alone as a magnifying glass is:

$M \approx \frac{N}{f_e} + 1$

For $f_e = 3.0$ cm, $N = 25$ cm: $M \approx 9.3$

Telescope

The astronomical telescope is designed to view distant objects. The objective lens forms a real image near its focal plane, which is then magnified by the eyepiece for comfortable viewing with a relaxed eye.

• Objective lens (fo): Large focal length, gathers light from distant objects.

• Eyepiece lens (fe): Short focal length, magnifies the image formed by the objective.

• Length of telescope (L): $L \approx f_o + f_e$

Angular magnification (M) of a telescope:

$M \approx -\frac{f_o}{f_e}$

• Example: $f_o = 985$ mm, $f_e = 5.00$ mm:

• $M = -\frac{985}{5.00} = -197$

• $L = 985 + 5.00 = 990$ mm = 99.0 cm

Wave Nature of Light: Interference and Diffraction

Snell's Law and Refraction

Snell's Law describes how light bends when passing from one medium to another:

$n_1 \sin \theta_1 = n_2 \sin \theta_2$

• $n$ is the index of refraction, $\theta$ is the angle with respect to the normal.

• When $n_2 > n_1$, light slows down and bends toward the normal.

Dispersion of Light: Prisms and Rainbows

Dispersion occurs because the index of refraction depends on wavelength. Shorter wavelengths (violet) are refracted more than longer wavelengths (red), causing white light to spread into a spectrum.

• Prisms separate light into its constituent colors due to wavelength-dependent refraction.

• Rainbows are formed by dispersion and internal reflection in water droplets.

Table: Indices of Refraction for Crown Glass

Approximate Color	Wavelength in Vacuum (nm)	Index of Refraction, n
Red	660	1.520
Orange	610	1.522
Yellow	580	1.523
Green	550	1.526
Blue	470	1.531
Violet	410	1.538

Diffraction

Diffraction is the bending of waves around obstacles or through openings. The extent of diffraction increases as the wavelength becomes comparable to the size of the opening or obstacle.

• For an opening of width $D$ and wavelength $\lambda$:

$\theta \approx \frac{\lambda}{D}$

• When $\lambda \sim D$, $\theta \sim 1$ radian ($\approx 57^\circ$).

Huygens' Principle

According to Huygens' principle, every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront is the tangent to all these wavelets.

Principle of Linear Superposition

When two or more light waves overlap, their electric fields add together. This can result in constructive (amplified) or destructive (diminished) interference, depending on their phase relationship.

• Constructive interference: Waves arrive in phase, amplitudes add.

• Destructive interference: Waves arrive out of phase, amplitudes subtract.

• Coherent sources: Maintain a constant phase relationship, necessary for stable interference patterns.

Young's Double-Slit Experiment

Young's experiment demonstrates the wave nature of light by producing an interference pattern of bright and dark fringes on a screen. Two slits act as coherent sources, and the pattern depends on the wavelength and slit separation.

• Bright fringes (constructive interference): $d \sin \theta = m \lambda$

• Dark fringes (destructive interference): $d \sin \theta = (m + \frac{1}{2}) \lambda$

• $d$ = slit separation, $\lambda$ = wavelength, $m$ = order of fringe (0, 1, 2, ...)

The distance $y$ from the central maximum to the $m$-th bright fringe on a screen at distance $L$ is:

$y = L \tan \theta \approx L \sin \theta$ (for small angles)

• Example: For $\lambda = 664$ nm, $d = 0.000120$ m, $L = 2.75$ m, third-order bright fringe:

• $\sin \theta = \frac{3 \times 664 \times 10^{-9}}{1.20 \times 10^{-4}} = 0.0166$

• $y = 2.75 \times 0.0166 = 0.0456$ m = 4.56 cm

White Light in Young's Experiment

When white light is used, each wavelength produces its own set of fringes. The central fringe is white (all colors overlap), while colored fringes appear on either side, with red farther from the center than green due to its longer wavelength.

Single-Slit Diffraction and Double-Slit Interference

Single-slit diffraction produces a central bright maximum with decreasing intensity maxima on either side. Double-slit interference patterns are superimposed on the single-slit envelope.

Summary Table: Comparison of Interference and Diffraction

Phenomenon	Cause	Pattern
Double-slit interference	Superposition of waves from two slits	Equally spaced bright and dark fringes
Single-slit diffraction	Bending of light through a single slit	Central maximum with decreasing side maxima

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