Optics

Light & Optics

Part 1 of 4 — Light, Reflection & Refraction

Light Is an Electromagnetic Wave

Light is oscillating electric and magnetic fields, perpendicular to each other and to the travel direction — a TRANSVERSE wave needing no medium. In vacuum every EM wave moves at

$c = 3 \times 10^8\;\text{m/s}, \quad c = f\lambda$

Visible light spans roughly $400\;\text{nm}$ (violet) to $700\;\text{nm}$ (red). Energy scales with frequency ($E = hf$): the spectrum runs radio → microwave → infrared → visible → ultraviolet → X-ray → gamma, in order of increasing frequency and energy.

Reflection

Angle of incidence equals angle of reflection, both measured from the NORMAL (the perpendicular to the surface), never from the surface itself:

$\theta_i = \theta_r$

Index of Refraction

Light slows down inside matter:

$n = \frac{c}{v} \geq 1$

(air $\approx 1.00$, water $1.33$, glass $\approx 1.5$, diamond $2.4$). Crossing a boundary, the FREQUENCY never changes — it is set by the source. The wavelength shrinks instead: $\lambda_{medium} = \lambda_{vacuum}/n$.

Snell's Law

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

• Into a DENSER medium ($n_2 > n_1$): the ray bends TOWARD the normal (slows down, straightens up).
• Into a less dense medium: bends AWAY from the normal.
• Hitting the boundary head-on ($\theta_1 = 0$): no bending at all, just a speed change.

Total Internal Reflection

Going from HIGH $n$ to LOW $n$, the refracted ray bends away from the normal — and at the critical angle it skims along the surface ($\theta_2 = 90^\circ$):

$\sin\theta_c = \frac{n_2}{n_1}$

Beyond $\theta_c$, no light escapes: it is ALL reflected back inside. Requirements: dense-to-rare travel AND incidence beyond critical. This traps light inside optical fibers — the physics of endoscopes and laparoscopic surgery.

Dispersion

$n$ varies slightly with wavelength (higher for violet than red in glass), so a prism fans white light into a spectrum — violet bent most, red least. Same physics paints the rainbow.

Worked Example — Snell's Law into Glass

A ray in air strikes a glass block ($n = 1.5$) at $30^\circ$ from the normal. Find the refraction angle.

Step 1 — Snell.

$(1.00)\sin 30^\circ = (1.5)\sin\theta_2$

Step 2 — Solve.

$\sin\theta_2 = \frac{0.5}{1.5} = 0.33 \quad\Rightarrow\quad \theta_2 \approx 19^\circ$

Bent toward the normal ($19^\circ < 30^\circ$), as entering a denser medium demands. Exiting the far parallel face, the ray bends back to $30^\circ$ — displaced sideways but parallel to its original path.

Worked Example — The Critical Angle of an Optical Fiber

Light travels inside glass ($n = 1.5$) toward a boundary with air. At what angles is it trapped?

Step 1 — Critical angle.

$\sin\theta_c = \frac{n_2}{n_1} = \frac{1.00}{1.5} = 0.67 \quad\Rightarrow\quad \theta_c \approx 42^\circ$

Step 2 — Interpret. Any ray meeting the wall at MORE than $42^\circ$ from the normal (a glancing hit) reflects totally, losing nothing; a fiber's gentle curves keep every bounce beyond critical, so the beam ricochets down kilometers of glass — or around the bends of an endoscope.

Step 3 — Discrimination check. Could light going from air INTO glass totally reflect? Never: entering a denser medium bends rays toward the normal, so the refracted ray always exists. TIR is strictly a dense-to-rare phenomenon.

Light, Reflection & Refraction 🎯

Key Takeaways — Part 1

• Light: transverse EM wave; $c = f\lambda$; energy up with frequency (radio lowest, gamma highest)
• Reflection: $\theta_i = \theta_r$, angles always from the normal
• $n = c/v$; crossing media, frequency fixed, wavelength and speed scale down by $n$
• Snell: $n_1\sin\theta_1 = n_2\sin\theta_2$; denser = toward normal
• TIR: dense to rare only, beyond $\sin\theta_c = n_2/n_1$ — fiber optics, endoscopes
• Dispersion: $n$ grows toward violet, so prisms bend violet most

Light & Optics

Part 2 of 4 — Mirrors & Image Formation

Plane Mirrors

A flat mirror forms an image that is virtual (no light actually passes through it), upright, the same size, and as far BEHIND the mirror as the object is in front. The image appears left-right reversed (what the mirror actually reverses is front-back, along its own axis — the "left-right swap" is how we describe facing our reversed self); up and down are not flipped.

Spherical Mirrors: the Cast

• Concave mirror (caves in toward the object): CONVERGING; focal length positive, $f = R/2 > 0$.
• Convex mirror (bulges out): DIVERGING; focal length negative, $f = R/2 < 0$.

The Mirror Equation and Magnification

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}, \quad m = -\frac{d_i}{d_o}$

Sign conventions (mirrors):

• $d_o > 0$ for a real object in front.
• $d_i > 0$: image IN FRONT of the mirror — real and (single mirror) inverted.
• $d_i < 0$: image BEHIND the mirror — virtual and upright.
• $|m| > 1$ enlarged, $|m| < 1$ reduced; $m < 0$ inverted, $m > 0$ upright.

The pairing never breaks for a single mirror or lens: real goes with inverted; virtual goes with upright.

Concave Mirror — Three Regimes

At exactly $d_o = 2f$: real, inverted, same size at $2f$. At $d_o = f$: no image (rays exit parallel — the searchlight configuration).

Convex Mirror — One Regime

Wherever the object stands, a convex mirror gives a virtual, upright, reduced image behind the mirror. The shrunken image buys a wide field of view: store-security mirrors and the passenger-side mirror ("objects are closer than they appear").

Worked Example — Concave Mirror, Object Outside the Center

An object stands $30\;\text{cm}$ from a concave mirror of focal length $10\;\text{cm}$.

Step 1 — Mirror equation.

$\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} = \frac{1}{10} - \frac{1}{30} = \frac{3}{30} - \frac{1}{30} = \frac{2}{30}$

so $d_i = 15\;\text{cm}$.

Step 2 — Classify. $d_i > 0$: a REAL image $15\;\text{cm}$ in front of the mirror, projectable onto a screen.

Step 3 — Magnification.

$m = -\frac{d_i}{d_o} = -\frac{15}{30} = -0.5$

Inverted, half size — matching the "beyond $C$" row of the table ($C = 2f = 20\;\text{cm}$).

Worked Example — Convex Mirror Sanity Check

The same object stands $20\;\text{cm}$ from a CONVEX mirror with $|f| = 20\;\text{cm}$ (so $f = -20\;\text{cm}$).

Step 1 — Mirror equation with signs.

$\frac{1}{d_i} = \frac{1}{-20} - \frac{1}{20} = -\frac{2}{20} = -\frac{1}{10}$

so $d_i = -10\;\text{cm}$: virtual, $10\;\text{cm}$ BEHIND the mirror.

Step 2 — Magnification.

$m = -\frac{(-10)}{20} = +0.5$

Upright, half size — the only kind of image a convex mirror can make. On the MCAT, if a convex mirror (or diverging lens) answer choice says "real" or "inverted," eliminate it without computing.

Mirrors & Image Formation 🎯

Key Takeaways — Part 2

• Plane mirror: virtual, upright, same size, equal distance behind
• $f = R/2$; concave converging ($f > 0$), convex diverging ($f < 0$)
• $1/f = 1/d_o + 1/d_i$, $m = -d_i/d_o$; $d_i > 0$ real and in front, $d_i < 0$ virtual and behind
• Real pairs with inverted; virtual pairs with upright — always, for single elements
• Concave: three regimes by object position; inside $F$ = upright magnifier
• Convex: always virtual, upright, reduced — wide field of view

Light & Optics

Part 3 of 4 — Thin Lenses, Diopters & the Eye

Lens Types

• Converging (convex, thicker in the middle): $f > 0$. Brings parallel rays to a real focus.
• Diverging (concave, thinner in the middle): $f < 0$. Spreads parallel rays as if from a virtual focus.

Note the vocabulary flip from mirrors: the CONVEX lens is the converging one.

Thin-Lens Equation

Same machinery as mirrors:

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}, \quad m = -\frac{d_i}{d_o}$

Sign convention for lenses: a REAL image ($d_i > 0$) forms on the far side of the lens from the object (light passes through), and is inverted. A virtual image ($d_i < 0$) appears on the object's own side, upright. The object-position regimes mirror the concave-mirror table exactly:

• $d_o > 2f$: real, inverted, reduced (the camera).
• $2f > d_o > f$: real, inverted, enlarged (the projector).
• $d_o < f$: virtual, upright, ENLARGED — the magnifying glass.
• Diverging lens: ALWAYS virtual, upright, reduced (like the convex mirror).

Lens Power — Diopters

$P = \frac{1}{f\;(\text{in meters})}$

in diopters (D). Strong lens = short focal length = high power. For thin lenses in CONTACT, powers simply add:

$P_{total} = P_1 + P_2 + \cdots$

This additivity is why prescriptions are written in diopters.

The Eye as a Lens System

Cornea plus crystalline lens (about $+60$ D combined, cornea doing most of it) form a REAL, INVERTED image on the retina; the brain re-inverts. Accommodation: ciliary muscles round the lens to add power for near objects.

• Myopia (nearsighted): eye too long or optics too strong — distant objects focus IN FRONT of the retina. Fix: DIVERGING lens (negative power) to back the focus up.
• Hyperopia (farsighted): eye too short — near objects focus BEHIND the retina. Fix: CONVERGING lens (positive power).
• Presbyopia: the aging lens stiffens and accommodation fades — reading glasses (converging), even for the emmetrope.

Memory anchor: the SIGN of the prescription names the problem — negative diopters = myopia, positive = hyperopia/presbyopia.

Worked Example — Projector Geometry

An object sits $15\;\text{cm}$ from a converging lens of focal length $10\;\text{cm}$.

Step 1 — Locate the image.

$\frac{1}{d_i} = \frac{1}{10} - \frac{1}{15} = \frac{3}{30} - \frac{2}{30} = \frac{1}{30}$

so $d_i = +30\;\text{cm}$: real, on the far side, projectable.

Step 2 — Magnification.

$m = -\frac{30}{15} = -2$

Inverted and doubled — the projector regime ($f < d_o < 2f$), which is why slides are loaded upside down.

Step 3 — Slide the object inside $f$. At $d_o = 5\;\text{cm}$: $\frac{1}{d_i} = \frac{1}{10} - \frac{1}{5} = -\frac{1}{10}$, so $d_i = -10\;\text{cm}$ and $m = +2$ — virtual, upright, doubled: the same lens is now a magnifying glass.

Worked Example — Writing a Prescription

A myopic patient's far point is $50\;\text{cm}$: anything beyond that blurs. What contact lens fixes distance vision?

Step 1 — The job. The lens must take an object at infinity and image it at the patient's far point, $50\;\text{cm}$ on the same side (virtual): $d_o = \infty$, $d_i = -0.5\;\text{m}$.

Step 2 — Thin lens.

$\frac{1}{f} = \frac{1}{\infty} + \frac{1}{-0.5} = -2\;\text{D}$

Step 3 — Read it. $P = -2\;\text{D}$ ($f = -0.5\;\text{m}$): a diverging lens, negative power — exactly the sign myopia demands. A $+2$ D answer choice is the trap for the farsighted case.

Thin Lenses, Diopters & the Eye 🎯

Key Takeaways — Part 3

• Convex lens converges ($f > 0$), concave diverges ($f < 0$) — vocabulary flipped from mirrors
• Same equations: $1/f = 1/d_o + 1/d_i$, $m = -d_i/d_o$; real images on the FAR side for lenses
• Regimes: beyond $2f$ camera, $f$ to $2f$ projector, inside $f$ magnifier; diverging = always virtual upright reduced
• $P = 1/f$ (meters) in diopters; contact lenses stack by adding powers
• Eye about $+60$ D; accommodation adds near power
• Myopia → minus (diverging) lenses; hyperopia/presbyopia → plus (converging)

Light & Optics

Part 4 of 4 — MCAT Integration: Wave Optics & Passage Strategy

Geometric optics treats light as rays; wave optics is what happens when light meets structures comparable to its wavelength. The MCAT tests both — and the judgment of which model applies.

Diffraction — Light Bends Around Edges

Passing through a slit, light spreads. The spreading is dramatic when the slit width is comparable to $\lambda$ and negligible when the opening is huge by comparison — why sound (meter wavelengths) bends around doorways but light (half a micron) seems to travel in straight lines. Diffraction sets the resolution limit of microscopes: you cannot resolve detail much smaller than the wavelength used. Electron microscopes win by using far shorter effective wavelengths.

Interference — Waves Add

Two coherent waves meeting:

• Path difference $= m\lambda$ ($m = 0, 1, 2, \ldots$): crest meets crest — constructive, bright.
• Path difference $= (m + \frac{1}{2})\lambda$: crest meets trough — destructive, dark.

Young's double slit (slit separation $d$, angle $\theta$ to a bright fringe):

$d\sin\theta = m\lambda \quad (\text{bright fringes})$

Longer wavelength or narrower slit spacing → wider fringe spread. The pattern is the proof that light is a wave; thin-film soap-bubble colors are the same interference with path difference set by film thickness.

Polarization — the Transverse Fingerprint

Only TRANSVERSE waves can be polarized (their oscillation has a sideways direction to filter). Polarizers pass one plane of oscillation; two polarizers crossed at $90^\circ$ pass nothing. Sound — longitudinal — can never be polarized. Optically active molecules (chiral sugars, amino acids) rotate the plane of polarized light: the bridge to your biochemistry deck.

Passage Strategy — the Optics Playbook

1. Choose the model first. Image locations and magnification: ray model, lens/mirror equation. Fringes, resolution limits, iridescence: wave model.
2. Signs before algebra. Diverging elements (convex mirror, concave lens): $f < 0$ and the image MUST be virtual, upright, reduced — eliminate contradicting choices without computing.
3. Real = inverted = opposite side (lens) / same side (mirror). Virtual = upright. Any answer choice pairing "real" with "upright" for a single element is wrong.
4. Multi-lens systems (microscope, telescope, corrective lens + eye): march left to right — the image of lens 1 is the OBJECT of lens 2. Total magnification multiplies: $m = m_1 m_2$.
5. Estimate with clean fractions. Optics arithmetic is reciprocals of small integers; get common denominators, never decimals.

Worked Example — Reading a Double-Slit Experiment (Passage Style)

Monochromatic light through slits $d = 2\;\mu\text{m}$ apart puts the first bright fringe ($m = 1$) at $\sin\theta = 0.25$. What is the wavelength — and what happens with red vs. blue light?

Step 1 — Solve the fringe condition.

$\lambda = \frac{d\sin\theta}{m} = (2 \times 10^{-6})(0.25) = 5 \times 10^{-7}\;\text{m} = 500\;\text{nm}$

Green — comfortably visible, a built-in sanity check.

Step 2 — Predict the color trend. $\sin\theta = m\lambda/d$: red (longer $\lambda$) pushes fringes FARTHER apart; blue pulls them tighter. White light gives a central white fringe with rainbow-edged side fringes, red outermost.

Step 3 — Knob check. Halving $d$ doubles $\sin\theta$ — narrower slit spacing spreads the pattern. Filling the apparatus with water ($\lambda$ shrinks by $n$) compresses it.

Worked Example — Two Lenses in Series

Light from an object $30\;\text{cm}$ left of lens 1 ($f_1 = +10\;\text{cm}$); lens 2 ($f_2 = +5\;\text{cm}$) stands $20\;\text{cm}$ right of lens 1.

Step 1 — Lens 1. $\frac{1}{d_i} = \frac{1}{10} - \frac{1}{30} = \frac{2}{30}$, so $d_i = +15\;\text{cm}$: real image $15\;\text{cm}$ right of lens 1, $m_1 = -\frac{15}{30} = -\frac{1}{2}$.

Step 2 — Feed lens 2. That image sits $20 - 15 = 5\;\text{cm}$ LEFT of lens 2: a real object at $d_o = 5\;\text{cm} = f_2$.

Step 3 — Lens 2. Object at the focal point → rays emerge PARALLEL; the final image is at infinity, ideal for a relaxed eye — this is how eyepieces are positioned in microscopes and telescopes.

MCAT Integration: Wave Optics 🎯

Key Takeaways — Part 4

• Ray model for images; wave model for fringes, resolution, and iridescence — choose before computing
• Diffraction matters when openings are of order $\lambda$; it caps microscope resolution near the wavelength
• Interference: path difference $m\lambda$ bright, $(m + \frac{1}{2})\lambda$ dark; double slit $d\sin\theta = m\lambda$
• Longer $\lambda$ or smaller $d$ = wider fringe pattern
• Polarization = transverse-only; crossed polarizers block all; chiral molecules rotate the plane
• Multi-lens systems: image of one is the object of the next; magnifications multiply; object at focus → image at infinity