Dispersion, Apparent Depth, and Optical Reasoning

Concept: n varies with wavelength. Different wavelengths travel at different speeds in materials. Because speed differences change bending amounts, colours spread out.

Concept: shorter wavelengths often have slightly higher n in glass. In many materials, shorter wavelengths experience slightly higher refractive index. That makes blue light slow slightly more and bend slightly more than red.

Concept: wavelength-dependent refraction separates colours. Each colour bends by a different amount as it enters and leaves the prism.

Concept: refraction shifts the apparent position upward. Refraction at the water–air surface shifts the apparent position upward.

In optics, the apparent depth of an object submerged in a medium appears shallower than its actual depth due to the bending of light rays at the interface between two different media. The relationship between apparent depth and real depth can be expressed as the apparent depth being equal to the real depth divided by the refractive index of the medium. This formula, \( \text{apparent depth} = \frac{\text{real depth}}{n} \), simplifies to \( d/n \) for small angles, where \( d \) represents real depth and \( n \) is the refractive index.

Concept: use d_app = d/n. d_app = 12/1.33 = 9.0 cm. The coin appears closer to the surface because rays bend away from the normal as they leave water.

Concept: frequency is fixed by the source. The boundary changes the wave’s speed in the medium, not the oscillation rate.

Concept: v=fλ with constant f. Lower v with constant f means smaller λ. So wavelength decreases when light enters a higher-n medium.

Concept: apply Snell’s law correctly. Snell’s law gives n_1 sin(θ_1)=n_2 sin(θ_2). Rearranging gives sin(θ_2)=(n_1/n_2)sin(θ_1).

Concept: use sine-angle matching. sin(28°)≈0.469, so θ_2≈28°. Because the ray enters glass (higher n), the refracted angle is smaller than 45°.

Concept: speed decrease leads to bending toward the normal. Slower speed corresponds to smaller refracted angle.

Concept: what optical boundary effects exist. a–c are optical boundary effects; fusion is unrelated.

Concept: TIR requires high-to-low refractive index. TIR requires light trying to go from higher n to lower n.

Concept: compute critical angle from sin(θ_c). sin(42°)≈0.67, so the critical angle is about 42°.

Optical fibres consist of a central core surrounded by a cladding layer. The core has a higher refractive index (n) than the cladding, which allows light to be trapped within the core through total internal reflection. This difference in refractive indices ensures that light signals can travel long distances with minimal loss, making optical fibres efficient for communication and data transmission. The cladding acts as a barrier, preventing light from escaping and maintaining the integrity of the signal.

Concept: gradual n changes bend light continuously. Gradients in refractive index bend light paths.

Concept: refraction can happen in a gradient, not just a sharp boundary. Temperature gradients create refractive index gradients.

Concept: compare incident angle to critical angle. 60° > critical angle → TIR.

Concept: bending away from the normal shifts the image upward. Your eye interprets the refracted rays as straight-line rays in air.

Concept: Snell’s law predicts angles from material properties. Snell’s law with n predicts bending angles.