Reflection, Refraction and Lenses

• Plane mirrors produce a virtual image (light rays do not actually pass through it, but appear to originate behind the mirror).
• The image is also upright, laterally inverted (left and right swapped), and the same distance behind the mirror as the object is in front.

Refraction at a Plane Surface

• If you place a perfectly straight pencil into a glass of water, it magically appears broken or bent at the surface.
• This is caused by refraction, which is the change in direction of a wave as it passes from one medium into another of different optical density.
• When a wavefront hits a boundary at an angle, one side of the wavefront changes speed before the other side, causing the entire wave to "swing" and change direction.
• To describe refraction in a step-by-step ray diagram:
  1. Draw the boundary and a dashed normal line at $90^\circ$.
  2. Draw the incident ray meeting the boundary.
  3. If moving into a denser medium (e.g., air to glass), light slows down and the refracted ray bends towards the normal (the angle of refraction, $r$, is smaller than $i$).
  4. If moving into a less dense medium (e.g., glass to air), light speeds up and the ray bends away from the normal ($r$ is greater than $i$).
• Note a crucial negative feature: a ray entering at exactly $90^\circ$ to the surface (along the normal) does NOT change direction, though its speed and wavelength still change.

Calculating Refractive Index

• The refractive index ($n$) represents how much a medium slows down light compared to its maximum speed in a vacuum.
• A higher refractive index means the material is more optically dense and light travels slower through it.

Where:

• $n$ = refractive index (no units)
• $c$ = speed of light in a vacuum ($3.00 \times 10^8\text{ m/s}$)
• $v$ = speed of light in the medium ($\text{m/s}$)

Dispersion Through a Prism

• Raindrops are just plain water, but they can split white sunlight into a brilliant rainbow.
• White light is a mixture of all the colours in the visible spectrum, and each colour travels at a slightly different speed inside a medium like glass or water.
• This causes dispersion, where white light separates into its constituent colours because they refract by different amounts.
• To describe dispersion through a prism step-by-step:
  1. Draw the incident ray of white light hitting the first boundary of the prism.
  2. Because violet light travels the slowest, it bends (refracts) the most towards the normal upon entry.
  3. Red light travels the fastest and bends the least, creating the top of the spectrum.
  4. At the second boundary (exiting the prism), refraction happens again, bending the rays away from the normal; violet bends furthest away, increasing the separation (the angle of deviation).
• The resulting spectrum always follows the order of ROYGBIV (Red to Violet).

Convex and Concave Lenses

• Understanding exactly how light bends allows us to correct human vision, turning a blurry world sharp using engineered pieces of glass.
• A convex lens is thicker in the middle and causes parallel rays of light to converge (come together).
• A concave lens is thinner in the middle and causes parallel rays to diverge (spread apart).
• Both lenses have a principal axis (a horizontal line through the optical centre) and a principal focus ($F$).
• The focal length ($f$) is the distance from the centre of the lens to the principal focus.
• For a convex lens, the properties of the image change depending on the object's distance from the lens:
  • Object $> 2F$: The image is real, inverted, and diminished.
  • Object $= 2F$: The image is real, inverted, and the same size.
  • Object between $F$ and $2F$: The image is real, inverted, and magnified.
  • Object $< F$ (Magnifying Glass): The image is virtual, upright, and magnified.
• In contrast, concave lenses always produce images that are virtual, upright, and diminished, regardless of the object's distance.

Drawing Lens Ray Diagrams

Follow these step-by-step rules to construct lens ray diagrams:

• For a Convex Lens:
  1. Draw the lens symbol (a vertical line with outward-pointing arrows $\uparrow$).
  2. Label the principal focus ($F$) and the $2F$ point (twice the focal length) on the principal axis on both sides of the lens for reference.
  3. Draw a ray from the top of the object parallel to the principal axis; it refracts through the principal focus ($F$) on the far side of the lens.
  4. Draw a second ray straight through the optical centre of the lens without bending.
  5. Where the rays meet, a real image is formed (unless the object is closer than $F$, where you must extend the rays backwards to form a virtual image).
• For a Concave Lens:
  1. Draw the lens symbol (a vertical line with V-shaped inward arrows).
  2. Draw a ray parallel to the principal axis; it refracts upwards, diverging so it appears to come from the principal focus ($F$) on the near side.
  3. Draw a dashed virtual line backwards to $F$.
  4. Draw a second ray straight through the optical centre.
  5. Where the dashed line and the straight ray cross, a virtual image is formed.
• Concave lenses are used to correct myopia (short-sightedness) and importantly, they do NOT form real images—their images are always virtual, upright, and diminished.

Wave Transmission, Speed, Frequency, and Wavelength

• When you hear muffled music from underwater in a swimming pool, why does the pitch sound exactly the same as in the air?
• The pitch is determined by the wave's frequency (measured in hertz (Hz)), and when transmission occurs across a boundary, the frequency remains absolutely constant.
• Waves cannot be created or destroyed at a boundary; the number of wave crests arriving per second must equal the number leaving per second.
• Because frequency ($f$) is constant, any change in wave speed ($v$) must result in a proportional change in wavelength ($\lambda$), according to the wave equation:

• If a wave slows down (e.g., light entering glass from air), its wavelength must decrease proportionally.
• Conversely, sound travels faster in solids than in gases because particles are closer together. If a sound wave moves from air into steel, its speed increases, so its wavelength must also increase.

Worked Example: Transmission Calculation

A $500\text{ Hz}$ sound wave moves from air ($v = 330\text{ m/s}$) to steel ($v = 5000\text{ m/s}$). Calculate the change in wavelength.

Step 1: Use the rearranged formula $\lambda = \frac{v}{f}$ for the air medium.

• $\lambda_{\text{air}} = \frac{330}{500} = 0.66\text{ m}$

Step 2: Use the formula for the steel medium (remembering frequency remains constant).

• $\lambda_{\text{steel}} = \frac{5000}{500} = 10\text{ m}$

Step 3: Calculate the difference to find the change in wavelength.

• Change in $\lambda = 10 - 0.66 = 9.34\text{ m}$