“Light: Reflection and Refraction”

🟢 Lite — Quick Review (1h–1d)

Rapid summary of reflection and refraction for last-minute NABTEB revision.

Reflection occurs when light bounces off a surface. Refraction occurs when light changes direction as it passes from one medium to another.

Laws of Reflection:

1. The angle of incidence ($i$) equals the angle of reflection ($r$): $i = r$
2. The incident ray, reflected ray, and normal all lie in the same plane

Key Definitions:

• Angle of incidence ($i$): Angle between incident ray and the normal
• Angle of reflection ($r$): Angle between reflected ray and the normal
• Normal: Line perpendicular to the reflecting surface at the point of incidence
• Virtual image: Image that cannot be formed on a screen (appears to be behind the mirror)

Types of Reflection:

• Regular (specular) reflection: From smooth surfaces like mirrors — rays remain parallel
• Diffuse reflection: From rough surfaces like paper — rays scatter in different directions

Mirror Formula: $$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$

Where $f$ = focal length, $u$ = object distance, $v$ = image distance. For concave mirrors, $f$ is positive; for convex mirrors, $f$ is negative.

Magnification: $$m = \frac{\text{image height}}{\text{object height}} = \frac{v}{u}$$

Laws of Refraction (Snell’s Law): $$\frac{\sin i}{\sin r} = \frac{n_2}{n_1} = \frac{v_2}{v_1}$$

Where $n$ is the refractive index, $i$ is angle of incidence in medium 1, $r$ is angle of refraction in medium 2.

Refractive Index: $$n = \frac{\text{speed of light in vacuum}}{\text{speed of light in medium}} = \frac{c}{v}$$

Absolute refractive index of a medium: $n = \frac{\sin i}{\sin r}$ (measured when light goes from vacuum to the medium).

Critical Angle: When light travels from a denser to a rarer medium: $\sin c = \frac{n_2}{n_1}$ (where $n_1 > n_2$)

Total Internal Reflection: Occurs when angle of incidence in the denser medium exceeds the critical angle. This is the principle behind fibre optic cables.

⚡ NABTEB Exam Tip: For mirror questions, use the sign conventions consistently — for real objects, object distance $u$ is always negative. For lenses, real objects have $u$ negative, real images have $v$ positive.

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🟡 Standard — Regular Study (2d–2mo)

For NABTEB students who want a solid understanding of reflection and refraction.

Plane Mirrors:

A plane mirror produces a virtual, laterally inverted image that is:

• The same size as the object
• The same distance behind the mirror as the object is in front
• Laterally inverted (left and right appear swapped)

Spherical Mirrors:

Concave mirrors (diverging):

• Bulge inward (curved away from incoming light)
• Reflective surface is on the inner, concave side
• Can produce real or virtual images depending on object position
• Used as shaving mirrors, dentist mirrors, and in torch reflectors
• Principal axis passes through centre of curvature (C) and pole (P); focal point (F) is midpoint of C and P, so $f = R/2$

Convex mirrors (diverging):

• Bulge outward
• Reflective surface is on the outer, convex side
• Always produce virtual, diminished (smaller) images
• Used as rear-view mirrors in vehicles (wider field of view)
• $f$ is taken as negative

Mirror Positions — Concave Mirror:

Object Position	Image Position	Image Type
Beyond C	Between C and F	Real, inverted, diminished
At C	At C	Real, inverted, same size
Between C and F	Beyond C	Real, inverted, magnified
At F	At infinity	No image formed (rays are parallel)
Between F and P	Behind mirror	Virtual, upright, magnified

Lens Formula: $$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$

Converging (Convex) Lens:

• Thicker in the middle
• Can produce real or virtual images
• Used in magnifying glasses, cameras, spectacles for hyperopia
• $f$ is positive

Diverging (Concave) Lens:

• Thinner in the middle
• Always produces virtual, diminished images
• Used in spectacles for myopia (nearsightedness)
• $f$ is negative

Lens Positions — Convex Lens:

Object Position	Image Position	Image Type
Beyond 2F	Between F and 2F	Real, inverted, diminished
At 2F	At 2F	Real, inverted, same size
Between 2F and F	Beyond 2F	Real, inverted, magnified
At F	At infinity	No real image formed
Between F and lens	Same side as object	Virtual, upright, magnified

Refraction Through a Glass Slab:

When light passes through a parallel-sided glass slab:

• It undergoes two refractions (air → glass, then glass → air)
• The emergent ray is parallel to the incident ray (lateral displacement occurs)
• The angle of deviation depends on the refractive index and angle of incidence

Dispersion:

White light splits into its component colours (spectrum) when passed through a prism because different wavelengths of light are refracted by different amounts. Violet light refracts most; red light refracts least.

$$n_{\text{violet}} > n_{\text{red}}$$

This is because the refractive index of a material is inversely proportional to wavelength.

Power of a Lens: $$P = \frac{1}{f} \text{ (in metres)}$$

Unit: Dioptre (D). Converging lenses have positive power; diverging lenses have negative power.

⚡ NABTEB Exam Tip: In refraction questions, always check whether light is going from rarer to denser (ray bends towards normal) or denser to rarer (ray bends away from normal).

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for NABTEB physics students who want thorough understanding.

Derivation of Mirror Formula:

For a concave mirror with object distance $u$ and image distance $v$:

From geometry of similar triangles (triangle formed by object height and image height): $$\frac{h’}{h} = \frac{v - f}{f} = \frac{-v}{u}$$

(negative sign because image is inverted relative to object)

So: $\frac{h’}{h} = -\frac{v}{u}$ and also $\frac{h’}{h} = \frac{v - f}{f}$

Equating: $-\frac{v}{u} = \frac{v - f}{f}$

Cross-multiplying: $-vf = uv - uf$

Rearranging: $uv = vf + uf = u(v + f)$

Dividing both sides by $uvf$: $$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$

Critical Angle and Total Internal Reflection Derivation:

When light goes from denser (medium 1, refractive index $n_1$) to rarer (medium 2, refractive index $n_2$): $$\frac{\sin i}{\sin r} = \frac{n_2}{n_1}$$

As $i$ increases, $r$ increases. At the critical angle $c$, $r = 90°$ (refracted ray grazes the surface): $$\sin c = \frac{n_2}{n_1}$$

Since $n_1 > n_2$, $\sin c < 1$, and $c$ has a real value. If $i > c$, $\sin r > 1$ — impossible, so no refraction occurs. This is total internal reflection.

Applications of Total Internal Reflection:

1. Optical fibres: Light signals travel through glass fibres by TIR, enabling internet and telephone communications
2. Prism periscopes: Right-angled prisms produce TIR, used in submarine periscopes
3. Diamond cutting: Diamond’s high refractive index (2.42) causes brilliant TIR, creating sparkle
4. Binoculars: Porro prisms use TIR to lengthen light path and produce erect images

Image Formation — Detailed Ray Diagrams:

For mirrors, two principal rays are sufficient to locate the image:

1. Ray parallel to principal axis — passes through F (concave) or appears to come from F (convex)
2. Ray through centre of curvature — reflects back on itself (C is on the path of the incident ray in spherical mirrors)
3. Ray through pole — reflects at equal angles with the normal

For lenses:

1. Ray parallel to axis — passes through F on the far side (convex) or diverges as if from F on the same side (concave)
2. Ray through optical centre — continues straight without deviation
3. Ray through focal point — emerges parallel to the axis

Refractive Index and Speed of Light:

Since $n = c/v$, light travels slower in denser media. This is because light interacts with atoms in the medium, causing a delay.

For light passing from medium $n_1$ to $n_2$: $$n_1 \sin i = n_2 \sin r$$

This is Snell’s Law in a form that does not require explicit velocity ratios.

Lens Maker’s Formula: $$\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

Where $R_1$ and $R_2$ are the radii of curvature of the two surfaces (positive if centre of curvature is on the outgoing side).

Combination of Thin Lenses:

For lenses in contact (or separated by a small distance): $$\frac{1}{f_{\text{eq}}} = \frac{1}{f_1} + \frac{1}{f_2} + \ldots$$

Power of combination: $P_{\text{eq}} = P_1 + P_2 + \ldots$

Aberrations:

1. Spherical aberration: Rays from the edge of a spherical mirror/lens focus at a different point than paraxial rays. Minimised by using parabolic mirrors or aperture stops.
2. Chromatic aberration: Different wavelengths focus at different points (dispersion). Corrected by using achromatic doublets (two lenses of different glasses).

⚡ NABTEB Quick Reference:

• Mirror formula: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$
• Lens formula: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$
• Magnification: $m = \frac{v}{u} = \frac{h’}{h}$
• Snell’s Law: $n_1 \sin i = n_2 \sin r$
• Critical angle: $\sin c = \frac{n_2}{n_1}$
• Power: $P = \frac{1}{f}$ (dioptres)
• Lens maker: $\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$