AP Physics 2: Geometric and Physical Optics

Optics in AP Physics 2 spans two complementary ways of understanding light. Geometric optics treats light as rays that travel in straight lines until they reflect or refract. It excels at predicting image location, magnification, and the behavior of mirrors and lenses. Physical optics treats light as a wave, which is essential for explaining interference and diffraction, phenomena that ray diagrams cannot capture.

A strong approach is to learn when each model applies, use the right equations with careful sign conventions, and connect math to what you would actually observe on a screen or in an instrument.

Geometric Optics: Reflection, Refraction, and Image Formation

Reflection and Mirrors

Reflection follows a simple rule:

• The angle of incidence equals the angle of reflection, measured from the normal.

For plane mirrors, the image is virtual, upright, and the same size as the object. The image appears the same distance behind the mirror as the object is in front.

For spherical mirrors (concave and convex), you often rely on the mirror equation and magnification:

$$\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}$$

$$m=\frac{h_i}{h_o}=-\frac{d_i}{d_o}$$

• $f$ is focal length.
• $d_o$ is object distance (typically positive for real objects in front of the mirror).
• $d_i$ is image distance (positive for real images in front of the mirror; negative for virtual images behind).
• $m$ is magnification; a negative value indicates an inverted image.

A concave mirror ($f>0$) can form real images (projectable on a screen) when the object is beyond the focal point. A convex mirror ($f<0$) always produces a virtual, upright, reduced image.

Practical insight: if you can project the image onto a screen, it is real, which usually corresponds to $d_i>0$ in standard conventions.

Refraction and Snell’s Law

Refraction occurs when light changes speed crossing a boundary between media with different indices of refraction $n$:

$$n_1\sin {\theta }_1=n_2\sin {\theta }_2$$

The index of refraction relates to light speed in the medium: $n=c/v$. A higher $n$ means a lower speed. When light enters a higher-index medium, it bends toward the normal; when it enters a lower-index medium, it bends away.

A key phenomenon is total internal reflection, which occurs only when light travels from higher $n$ to lower $n$ and the incidence angle exceeds the critical angle:

$$\sin {\theta }_c=\frac{n_2}{n_1}(n_1>n_2)$$

Total internal reflection is the basis of fiber optics and can also explain why submerged objects can appear distorted near the surface.

Lenses and the Thin Lens Equation

Thin lenses are treated with the same form as mirrors:

$$\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}$$

• Converging lenses have $f>0$ and can form real images for objects beyond the focal length.
• Diverging lenses have $f<0$ and form virtual images for real objects.

Magnification still follows:

$$m=\frac{h_i}{h_o}=-\frac{d_i}{d_o}$$

Ray diagrams help you interpret what the equations mean. For a converging lens, three principal rays are commonly used: a ray parallel to the axis refracts through the focal point, a ray through the center continues straight, and a ray through the focal point exits parallel. Diverging lenses use similar rays, but refracted rays diverge as if coming from a focal point on the object side.

Example reasoning (without plugging numbers): if an object is placed between a converging lens and its focal point, the lens cannot bring rays to a real focus on the far side. The image must be virtual ($d_i<0$), upright ($m>0$), and magnified.

Combining Lenses and Optical Power

In many practical systems, lenses are combined. A useful quantity is optical power $P$ measured in diopters:

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

For thin lenses in contact, powers add:

$$P_{\text{eq}}=P_1+P_2$$

This is widely used in eyeglasses and basic instrument design. If you see a system described in diopters, you can translate quickly to focal length and predict image behavior.

Physical Optics: Interference and Diffraction

Geometric optics predicts where rays go, but it cannot explain bright and dark fringe patterns. Those require the wave model, where phase differences lead to constructive or destructive interference.

Double-Slit Interference

In the classic double-slit experiment, coherent light passes through two narrow slits separated by distance $d$, forming an interference pattern on a screen a distance $L$ away. For small angles, bright fringes occur when the path difference is an integer multiple of wavelength:

$$d\sin \theta =m\lambda (m=0,\pm 1,\pm 2,\dots )$$

Dark fringes occur at half-integer path differences:

$$d\sin \theta =\left(m+\frac{1}{2}\right)\lambda$$

If the screen geometry is used with small-angle approximations, the fringe spacing $\Delta y$ is often expressed as:

$$\Delta y=\frac{\lambda L}{d}$$

Interpretation matters: decreasing slit separation makes fringes spread out, while increasing wavelength also spreads fringes. That is why red light (larger $\lambda$) produces wider spacing than blue light under the same setup.

Single-Slit Diffraction

Even one slit produces a pattern because different parts of the slit act as sources that interfere with each other. The central maximum is broad, and minima occur at:

$$a\sin \theta =m\lambda (m=1,2,3,\dots )$$

where $a$ is the slit width. A narrower slit produces a wider diffraction pattern. This is a central idea in optical resolution: restricting an aperture reduces light spread in ray terms, but increases spreading due to diffraction.

Interference Meets Diffraction

In many lab setups, you see both effects together. A double-slit interference pattern is often “wrapped” inside a single-slit diffraction envelope because each slit has finite width. The interference sets fine fringe spacing; diffraction controls overall intensity distribution.

This combination is not just a detail; it reflects the physical reality that no slit is infinitely thin, and real optical elements have finite apertures.

Key Connections and Exam-Ready Thinking

When to Use Rays vs Waves

• Use ray optics for image distance, magnification, and qualitative mirror or lens behavior.
• Use wave optics for patterns on a screen, conditions for maxima and minima, and how geometry affects fringe spacing and spread.

A common mistake is trying to force a ray diagram to explain dark fringes. Darkness in interference is not a lack of rays; it is cancellation due to phase.

Image Type, Orientation, and Signs

Be explicit about what you expect before calculating:

• Real images are typically inverted and can be projected.
• Virtual images are upright and cannot be projected onto a screen in the usual way.

Then let the sign of $d_i$ and the sign of magnification confirm that expectation. That habit prevents algebra from overriding physical sense.

What Optics Looks Like in the Real World

Reflection and refraction govern everyday perception: mirrors, windows, and apparent depth in water. Lenses underpin cameras, microscopes, and vision correction. Interference and diffraction explain thin-film colors, diffraction gratings in spectrometers, and the resolution limits of telescopes and microscopes. AP Physics 2 optics ties these observations to a small set of powerful principles, with each equation serving as a compact statement about what light must do under a given geometry.

Mastering both geometric and physical optics is less about memorizing formulas and more about predicting what should happen, then using math to quantify it. That is exactly the skill the course and exam are designed to reward.