Fundamental Optics Formulas

Why This Matters

Geometric optics is where physics gets visual—you're learning to predict exactly where images form, whether they're magnified or inverted, and why light bends or bounces the way it does. These formulas aren't just equations to memorize; they represent the ray model of light in action, showing how light travels in straight lines and interacts with boundaries between media. You'll be tested on your ability to apply these relationships to mirrors, lenses, and interfaces, connecting mathematical predictions to physical ray diagrams.

The key concepts here include refraction and Snell's law, image formation through the thin lens and mirror equations, magnification and image characteristics, and wave behavior including interference. Don't just memorize each formula—know what physical principle each one demonstrates. Can you explain why total internal reflection only happens when light moves from a denser to a less dense medium? Can you predict whether an image will be real or virtual based on sign conventions? That's the level of understanding AP Physics 2 demands.

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Refraction and Light Speed in Media

When light crosses a boundary between two materials, it changes speed—and that speed change causes bending. The index of refraction quantifies how much slower light travels in a medium compared to vacuum, and Snell's law describes the geometric consequence of that speed change.

Index of Refraction

• $n = c/v$—the ratio of light's speed in vacuum ($c$) to its speed in the medium ($v$)
• Higher $n$ values indicate light travels slower and bends more toward the normal when entering that medium
• Dimensionless quantity that characterizes optical density; $n = 1$ for vacuum, approximately $1.5$ for glass

Snell's Law

• $n_1 \sin\theta_1 = n_2 \sin\theta_2$—relates angles of incidence and refraction at a boundary between two media
• $\theta_1$ is the angle of incidence, $\theta_2$ is the angle of refraction, both measured from the surface normal
• Light bends toward the normal when entering a medium with higher $n$, away from normal when entering lower $n$

Critical Angle

• $\sin\theta_c = n_2/n_1$ (valid only when $n_2 < n_1$)—the incident angle beyond which total internal reflection occurs
• Total internal reflection happens only when light travels from a denser medium to a less dense medium
• Foundation of fiber optics—light stays trapped inside the fiber because it always exceeds the critical angle

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Image Formation: Lenses and Mirrors

The thin lens and mirror equations share identical mathematical forms because they describe the same geometric relationship between object position, image position, and focal length. The key difference lies in sign conventions and whether the optical element transmits or reflects light.

Thin Lens Equation

• $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$—relates focal length ($f$) to object distance ($d_o$) and image distance ($d_i$)
• Converging lenses have positive $f$, diverging lenses have negative $f$
• Positive $d_i$ indicates a real image on the opposite side of the lens from the object; negative $d_i$ means virtual image on the same side

Mirror Equation

• $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$—mathematically identical to the thin lens equation but applied to spherical mirrors
• Concave mirrors have positive $f$, convex mirrors have negative $f$
• Real images form in front of the mirror (positive $d_i$); virtual images form behind it (negative $d_i$)

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Magnification and Image Characteristics

Magnification tells you two things at once: how large the image is relative to the object, and whether it's upright or inverted. The sign of magnification encodes orientation, while the magnitude encodes size.

Lens Magnification

• $M = -d_i/d_o = h_i/h_o$—ratio of image height to object height, or negative ratio of distances
• Negative $M$ indicates an inverted (real) image; positive $M$ indicates an upright (virtual) image
• $|M| > 1$ means the image is enlarged; $|M| < 1$ means it's reduced

Mirror Magnification

• $M = -d_i/d_o = h_i/h_o$—identical formula to lens magnification
• Real images from concave mirrors are inverted (negative $M$); virtual images are upright (positive $M$)
• Convex mirrors always produce virtual, upright, reduced images ($M$ positive, $|M| < 1$)

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Wave Behavior and Interference

Light behaves as a wave when interference and diffraction effects become significant. These formulas connect wavelength, frequency, and geometry to predict where constructive and destructive interference occur.

Wave Equation

• $v = f\lambda$—relates wave speed ($v$) to frequency ($f$) and wavelength ($\lambda$)
• Frequency remains constant when light enters a new medium; wavelength changes as speed changes
• Fundamental relationship for all waves—connects the temporal property (frequency) to the spatial property (wavelength)

Diffraction Grating Equation

• $d\sin\theta = m\lambda$—condition for constructive interference maxima in a diffraction grating
• $d$ is the slit spacing, $m$ is the order number (0, 1, 2, ...), and $\theta$ is the angle to the maximum
• Higher orders (larger $m$) appear at larger angles; used to separate light into its component wavelengths

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Polarization and Special Angles

Brewster's angle represents a special case where reflected and refracted rays are perpendicular, resulting in perfectly polarized reflected light. This connects geometric optics to the wave nature of light.

Brewster's Angle

• $\tan\theta_B = n_2/n_1$—the angle of incidence at which reflected light is completely polarized
• Reflected ray is perpendicular to refracted ray at Brewster's angle, so reflected light contains only one polarization component
• Practical applications include reducing glare in photography and designing polarizing filters

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Quick Reference Table

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Self-Check Questions

1. Both the thin lens equation and mirror equation have the same mathematical form. What determines whether you use positive or negative values for $f$ and $d_i$ in each case?

2. Which two formulas both involve the ratio $n_2/n_1$, and what physically different phenomena do they describe?

3. A student calculates $M = -2.5$ for an image. What three things can you immediately conclude about this image?

4. Compare and contrast what happens to light's frequency, wavelength, and speed when it passes from air into glass. Which formula would you use to find the new wavelength?

5. An FRQ shows light traveling from water ($n = 1.33$) into air ($n = 1.00$) and asks whether total internal reflection is possible. How would you determine this, and what formula would you use to find the critical angle?