Geometrical Optics

In a homogeneous medium — one with uniform composition — light travels in a straight line, a behavior called rectilinear propagation. But when light reaches a boundary between two different materials, like air and water, its path can change. Geometrical optics is the study of how light interacts with surfaces at these boundaries, covering reflection, refraction, and the mirrors and lenses built from them.

Key Takeaways

• Reflection follows the law of reflection (angle of incidence = angle of reflection); images are real (rays converge) or virtual (rays only appear to converge).

• Plane mirrors always produce virtual, upright images; spherical mirrors (concave/convex) follow the mirror equation (1/f = 1/o + 1/i = 2/r) and magnification formula (m = −i/o).

• Refraction bends light at a media boundary per the index of refraction (n = c/v) and Snell's Law (n₁sinθ₁ = n₂sinθ₂); exceeding the critical angle (θc = sin⁻¹(n₂/n₁)) causes total internal reflection.

• Lenses refract light twice and follow the same equation forms as mirrors; real lenses use the lensmaker's equation, and lens power (P = 1/f, in diopters) adds across multiple lens systems.

• Spherical aberration and chromatic aberration both distort images formed by lenses and mirrors.

Reflection

Reflection happens when light strikes a surface and bounces back into its original medium instead of passing through.

The Law of Reflection

The law of reflection states that the angle of incidence (where light strikes the surface) equals the angle of reflection (where it bounces off). Both angles are measured relative to the normal, an imaginary line perpendicular to the surface at the point of contact.

Real vs. Virtual Images

Reflection is central to how mirrors form images, and every image a mirror produces is classified as one of two types:

• Real image: light rays actually converge at a point. If you placed a screen there, you'd see the image projected on it.

• Virtual image: light rays only appear to originate from a point — they don't actually converge there.

Plane Mirrors

A plane mirror has no curvature, so it doesn't converge or diverge reflected light — parallel rays stay parallel after reflection. Because the rays never converge, plane mirrors always produce virtual, upright images: the image appears to sit the same distance behind the mirror as the object sits in front of it, even though no real light rays pass through that point behind the mirror.

A useful way to think about a plane mirror: it's a special case of a spherical mirror with an infinite radius of curvature — essentially a flat segment of what would be an infinitely large sphere.

Spherical Mirrors: Concave and Convex

Spherical mirrors come in two varieties:

• Concave mirror: curves inward, like the inside of a sphere, causing light rays to converge.

• Convex mirror: curves outward, like the outside of a sphere, causing light rays to diverge.

Several key terms describe a spherical mirror's geometry:

• Center of curvature (C): the center of the imaginary sphere the mirror segment is taken from, located on the optical axis at a distance equal to the mirror's radius of curvature.

• Radius of curvature (r): the distance from the mirror's surface to the center of curvature.

• Focal point (F): the point on the optical axis where parallel rays converge (concave) or appear to diverge from (convex), positioned halfway between the mirror and the center of curvature.

• Focal length (f): the distance from the mirror to the focal point.

• Principal axis: the line passing through the center of curvature and the midpoint of the mirror.

• Object distance (o): the distance from the object to the mirror.

• Image distance (i): the distance from the resulting image to the mirror.

• Magnification (m): the ratio of image height to object height.

The Mirror Equation and Magnification

These quantities are linked by the mirror equation:

1/f = 1/o + 1/i = 2/r

If the image distance i comes out positive, the image is real and in front of the mirror; if negative, it's virtual and behind the mirror.

Magnification is calculated as:

m = −i/o

• A positive m (m > 0) means an upright image; a negative m (m < 0) means an inverted image.

• If |m| < 1, the image is smaller than the object; if |m| > 1, it's larger; if m = 1, they're the same size.

Worked example. An object sits 30 cm in front of a concave mirror with a focal length of 10 cm. Solving the mirror equation for image distance: 1/10 = 1/30 + 1/i, so 1/i = 1/10 − 1/30 = 2/30, giving i = 15 cm.

Since i is positive, the image is real, 15 cm in front of the mirror. Magnification: m = −i/o = −15/30 = −0.5, so the image is inverted and half the object's size.

Ray Diagram Scenarios

Ray diagrams trace the paths of light rays reflecting off a mirror's surface, approximating where an image forms, its size, and its orientation.

For a convex mirror, incoming parallel rays always diverge after reflection, so they never meet in front of the mirror — they only appear to originate from a point behind it. Convex mirrors therefore always produce a virtual, upright, reduced image, which is why they're used for vehicle side mirrors: a wider field of view with a smaller, upright image.

A concave mirror's image depends on where the object sits relative to the focal point (F) and center of curvature (C):

Concave mirror image formation by object position

Object Position	Image Type	Orientation	Size	Image Location
Beyond C	Real	Inverted	Reduced	Between F and C
At C	Real	Inverted	Same size	At C
Between C and F	Real	Inverted	Enlarged	Beyond C
At F	No image forms — reflected rays are parallel	—	—	—
Between F and mirror	Virtual	Upright	Enlarged	Behind the mirror

Sign Conventions for Mirrors

Quantity	Positive	Negative
Object distance (o)	Object in front of mirror	Object behind mirror (uncommon)
Image distance (i)	Real image, in front of mirror	Virtual image, behind mirror
Radius of curvature (r) / Focal length (f)	Concave mirror	Convex mirror
Magnification (m)	Upright image	Inverted image

Refraction

Refraction is the bending of light as it passes from one medium into another and changes speed.

Index of Refraction and Snell's Law

Light travels fastest in a vacuum, at c ≈ 3.00×10⁸ m/s. In any other medium, it slows down. The index of refraction (n) quantifies this:

n = c/v

where v is the speed of light in that medium. A higher index of refraction means light moves more slowly through the medium.

When light crosses a boundary between two media, this speed change bends its path according to Snell's Law:

n₁sinθ₁ = n₂sinθ₂

where n₁ and n₂ are the indices of refraction of the first and second media, and θ₁ and θ₂ are the angles of incidence and refraction (both measured from the normal).

• Entering a medium with a higher index of refraction (n₂ > n₁): light bends toward the normal.

• Entering a medium with a lower index of refraction: light bends away from the normal.

Critical Angle and Total Internal Reflection

When light travels from a higher-index medium into a lower-index medium (e.g., water to air), there's a specific angle of incidence at which the refracted ray skims exactly along the boundary rather than entering the second medium. This is the critical angle (θc):

θc = sin⁻¹(n₂/n₁)

where n₁ is the higher-index (initial) medium and n₂ is the lower-index (second) medium.

If the angle of incidence exceeds the critical angle, none of the light refracts into the second medium — all of it reflects back into the original medium. This is total internal reflection, the principle behind fiber optics and many optical instruments.

Lenses

Unlike mirrors, which reflect light, lenses work by refracting it — twice. Light refracts once entering the lens from air, and again exiting the lens back into air. This double refraction lets lenses focus or spread out light to form images.

Thin Spherical Lenses

A thin spherical lens has two focal points, one on each side, due to its symmetry. Thin lenses follow the same equation forms as mirrors:

1/f = 1/o + 1/i and m = −i/o

As with mirrors, a positive magnification means an upright image; a negative magnification means an inverted image.

Real Lenses and the Lensmaker's Equation

For lenses where thickness can't be ignored ("real lenses"), the focal length depends on the lens material's refractive index and the curvature of both surfaces. This relationship is the lensmaker's equation:

1/f = (n − 1)(1/r₁ − 1/r₂)

where f is the focal length, n is the refractive index of the lens material, and r₁ and r₂ are the radii of curvature of the lens's two surfaces (positive for a surface curving outward/convex, negative for a surface curving inward/concave).

The lensmaker's equation is named and its inputs are described in the source material, but the equation itself was shown only in an on-screen graphic that isn't captured in the transcript text. The form given here is the standard, independently verified lensmaker's equation for a thin lens.

Sign Conventions for Lenses

Quantity	Positive	Negative
Object distance (o)	Object on the same side of the lens as the light source	Object on the opposite side of the light source
Image distance (i)	Image on the opposite side of the lens from the light source (real image)	Image on the same side as the light source (virtual image)
Radius of curvature (r) / Focal length (f)	Convex (converging) lens	Concave (diverging) lens
Magnification (m)	Upright image	Inverted image

Lens Power

Lens power (P) is the inverse of focal length, measured in diopters (D):

P = 1/f (f in meters)

A positive power indicates a converging (convex) lens; a negative power indicates a diverging (concave) lens. Optometrists prescribe corrective lenses in diopters based on how much correction is needed to focus light onto the retina.

Multiple Lens Systems

When two or more lenses sit in contact or at negligible distance from each other, they act as a single combined system:

1/f = 1/f₁ + 1/f₂ + ⋯, P = P₁ + P₂ + ⋯, m = m₁ × m₂ × ⋯

The equivalent focal length and power add; the total magnification is the product of each lens's individual magnification.

As with the lensmaker's equation above, these three relationships are described in words in the source material, with the equations themselves shown only as on-screen graphics. The forms given here are the standard, independently verified relationships for lenses in contact — each follows directly from the same 1/f = 1/o + 1/i lens equation already established above.

Aberrations

Both lenses and mirrors can produce imperfect images due to aberrations:

• Spherical aberration: light rays passing through the edges of a lens or mirror focus at a different point than rays passing through the center, producing a blurred image.

• Chromatic aberration: different wavelengths of light refract at slightly different angles passing through a lens (dispersion), producing colored fringes around the image.

Common MCAT Mistakes

• Assuming a positive image distance always means "virtual." For mirrors, it's the opposite: a positive i means the image is real and in front of the mirror; a negative i means virtual and behind it.

• Treating a plane mirror as capable of forming a real image. A flat mirror never converges light, so it only ever produces a virtual, upright image — regardless of how far the object sits.

• Applying the critical angle formula in the wrong direction. Total internal reflection only happens going from a higher-index medium into a lower-index one; there's no critical angle for light entering a higher-index medium.

• Reading magnification sign as a size cue instead of an orientation cue. The sign of m tells you upright (positive) vs. inverted (negative) — it's the magnitude, |m|, that tells you whether the image is larger, smaller, or the same size.

MCAT-Style Concept Check

Question: A light ray traveling through glass (n = 1.5) strikes the boundary with air (n = 1.0) at an angle of incidence greater than the critical angle. What happens to the ray?

• A) It refracts into the air, bending away from the normal.

• B) It totally internally reflects back into the glass.

• C) It passes straight through the boundary undeviated.

• D) It splits evenly between reflection and refraction into the air.

Answer: B

Explanation: The ray is traveling from a higher-index medium (glass, n = 1.5) into a lower-index medium (air, n = 1.0), so a critical angle exists. Once the angle of incidence exceeds that critical angle, none of the light can refract into the air — all of it reflects back into the glass. This is total internal reflection, the same phenomenon that keeps light traveling down a fiber-optic cable.

FAQ

What is the difference between a real image and a virtual image?

A real image forms where light rays actually converge — if you placed a screen at that point, the image would appear on it. A virtual image forms where light rays only appear to originate from, based on their reflected or refracted paths; no light actually passes through that point, so it can't be projected on a screen.

What does a negative magnification mean?

A negative magnification (m < 0) means the image is inverted relative to the object. The sign of m only describes orientation — upright (positive) or inverted (negative) — not size; the magnitude, |m|, separately tells you whether the image is enlarged, reduced, or the same size as the object.

What is total internal reflection used for?

Total internal reflection occurs when light traveling from a higher-index medium to a lower-index medium hits the boundary at an angle beyond the critical angle, causing all of the light to reflect back rather than refract out. It's the principle behind fiber optics, where light stays trapped inside the fiber core as it travels.

How are lens power and focal length related?

Lens power (P) is the reciprocal of focal length in meters: P = 1/f, measured in diopters (D). A converging (convex) lens has positive power; a diverging (concave) lens has negative power. When multiple lenses sit in contact, their powers simply add.