When does a convex lens form a real versus virtual image?

A convex (converging) lens forms a real image when the object distance d_o is greater than the focal length f. The image is real, inverted, and appears on the opposite side of the lens from the object. When d_o < f, the lens cannot converge the rays quickly enough and the image is virtual, erect, and magnified � appearing on the same side as the object. When d_o = f the image is at infinity. The thin-lens equation 1/f = 1/d_o + 1/d_i describes all three cases: positive d_i means real image, negative means virtual.

How does a compound microscope produce high magnification?

A compound microscope uses two converging lenses in series. The objective lens (very short focal length, typically 2�20 mm) is placed just beyond its focal length from the specimen, forming a highly magnified real intermediate image inside the microscope tube. The eyepiece lens (10�25 mm focal length) then acts as a magnifying glass for the eye, further magnifying this intermediate image. Total magnification M � m_obj � M_eye, where m_obj = -d_i/d_o (typically -40 to -100�) and M_eye = 250/f_eye (typically 10�), giving total magnification of 400�1000�.

What causes chromatic aberration and how is it corrected?

Chromatic aberration arises because the refractive index of glass depends on wavelength (dispersion): n(?). Blue light bends more than red light, so a simple lens has a shorter focal length for blue than for red � different colours focus at different positions, producing colour fringes around images. The Abbe number V = (n_D-1)/(n_F-n_C) quantifies a glass's dispersion. Correction uses an achromatic doublet: a converging crown glass element (high V) cemented with a diverging flint glass element (low V). The doublet brings two wavelengths (typically red and blue) to the same focus while partially correcting the intermediate wavelengths. An apochromat corrects three wavelengths.

Optics Simulator � Lenses, Microscope, Telescope & Aberrations

Optics Physics

Thin-Lens Equation

All six presets use the paraxial (small-angle) approximation, where the thin-lens equation exactly governs image formation for a single thin lens:

Thin-lens equation: 1/f = 1/d_o + 1/d_i Solving for image distance: d_i = f�d_o / (d_o - f) Linear magnification: m = -d_i / d_o = h_i / h_o Sign conventions (Cartesian): Distances measured from lens plane Object left of lens: d_o > 0 Real image right: d_i > 0 Inverted (m < 0) Virtual image left: d_i < 0 Erect (m > 0) Converging lens: f > 0 Diverging lens: f < 0 (always virtual image)

Image Formation Cases

Object position	d_i	m	Image type
d_o > 2f	f < d_i < 2f (+ side)	\|m\| < 1	Real, inverted, diminished
d_o = 2f	d_i = 2f	m = -1	Real, inverted, same size
f < d_o < 2f	d_i > 2f (+ side)	\|m\| > 1	Real, inverted, magnified
d_o = f	d_i ? 8	�	Image at infinity (parallel beam out)
d_o < f	d_i < 0 (same side)	m > 1	Virtual, erect, magnified
Any d_o (concave)	d_i < 0 (always)	0 < m < 1	Virtual, erect, diminished

Compound Optical Instruments

Compound Microscope: M_total = m_obj � M_eye m_obj = -d_i / d_o (objective, d_o just > f_obj) M_eye = 250 / f_eye (standard near point = 250 mm) Typical: m_obj � -40, M_eye � 10 ? M = -400� Astronomical Telescope (afocal system): M_angular = -f_obj / f_eye (negative = inverted) Angular resolution: ?_min = 1.22�?/D (Rayleigh criterion) f_obj � f_eye ? tube length = f_obj + f_eye Lensmaker's Equation (thick lens): 1/f = (n-1)�[1/R1 - 1/R2 + (n-1)�d/(n�R1�R2)] For thin lens: d ? 0 ? 1/f = (n-1)�[1/R1 - 1/R2]

Aberrations

Chromatic aberration: Caused by dispersion: n = n(?) ? f = f(?) Abbe number: V = (n_D - 1)/(n_F - n_C) (F=486nm, D=589nm, C=656nm) Crown glass: V � 60 (low dispersion) Flint glass: V � 36 (high dispersion) Achromatic doublet: brings ?_red and ?_blue to same focus Power of doublet: f1/V1 + f2/V2 = 0 Spherical aberration: Marginal rays (large h) focus closer than paraxial rays Longitudinal SA � h�/(2f) for a plano-convex lens Correction: aspherical surfaces, aplanatic design, apochromats

Preset Guide

Preset	Key Physics	What to observe
🔍 Convex Lens	1/f=1/do+1/di; d_i>0 when d_o>f	Drag d_o slider: image flips at d_o=f from real?virtual
🔎 Concave Lens	f<0; d_i always negative ? virtual	Image always erect, smaller; \|m\|<1 at all object distances
🔬 Lens Combo	1/f_eff = 1/f1+1/f2-d/f1f2	Intermediate image feeds L2; overall m = product of individual m
🔬 Microscope	M = m_obj � 250/f_eye	Short f_obj ? large m_obj; observe total magnification product
🔭 Telescope	M_ang = f_obj/f_eye	Long f_obj with short f_eye; parallel rays in, angular magnification out
🌕 Aberrations	n(?) ? f(?); marginal ? paraxial	Different colours focus at different positions (chromatic); dashed marginal ray shorter focus (spherical)

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Curriculum Links

Level	Topic	Covered
GCSE Physics	Light, reflection and refraction	Convex/concave lenses, real/virtual image, ray diagrams
A-Level Physics	Optics, refraction, optical instruments	Thin-lens equation, magnification, microscope, telescope
AP Physics 2	Geometric optics, wave optics	1/f=1/do+1/di, lensmaker's equation, diffraction
IB Physics	Wave phenomena, optics	Image construction, compound instruments, resolution limit
University Physics	Geometrical and physical optics	Seidel aberrations, Rayleigh criterion, Fourier optics

Optics Simulator #86