Light: Laws of Reflection and Refraction

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your NECO exam.

Law of Reflection: The angle of incidence ($i$) equals the angle of reflection ($r$). Both angles are measured from the normal (perpendicular) to the reflecting surface. $$i = r$$

Law of Refraction (Snell’s Law): When light passes from one medium to another, $\dfrac{\sin i}{\sin r} = \dfrac{v_1}{v_2} = \dfrac{n_2}{n_1}$, or: $$n_1 \sin i = n_2 \sin r$$ where $n$ is the refractive index.

Refractive Index: $$n = \frac{c}{v} = \frac{\text{speed of light in vacuum}}{\text{speed of light in medium}}$$ Also: $n = \frac{\sin i}{\sin r}$ (for light going from air into a medium).

Critical Angle and Total Internal Reflection: When light goes from a denser to a rarer medium, if the angle of incidence exceeds the critical angle $c$, total internal reflection occurs. $$\sin c = \frac{n_2}{n_1} \quad \text{(where } n_1 > n_2\text{)}$$ Total internal reflection is used in fibre optic cables.

⚡ NECO Tip: Draw a clear diagram with the normal line drawn at $90°$ to the surface. Always measure angles FROM the normal. In NECO questions, watch for whether light is going from air → glass (refracted ray bends towards normal) or glass → air (refracted ray bends away from normal).

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🟡 Standard — Regular Study (2d–2mo)

Standard content for NECO Physics students with a few days to months.

Types of Reflection

• Regular (specular) reflection: Occurs on smooth surfaces like mirrors — all reflected rays are parallel. Image is clear and well-defined.
• Diffuse reflection: Occurs on rough surfaces — reflected rays scatter in different directions. This is why we can see most objects.

Image Formation by Plane Mirrors

• Image is the same distance behind the mirror as the object is in front
• Image is laterally inverted (left and right appear swapped)
• Image is virtual (cannot be projected on a screen)
• Image size = object size (magnification $m = 1$)

Spherical Mirrors

For concave mirrors (sign convention using real-is-positive): $$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$ where $f$ = focal length, $u$ = object distance, $v$ = image distance.

• Concave mirror: $f$ is positive (real focus). Used as a shaving mirror or in torch reflectors.
• Convex mirror: $f$ is negative. Used as a rear-view mirror — gives a wide field of view but smaller image.

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

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

Refraction Through Prisms

For a triangular prism: $$\mu = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$ where $A$ = angle of prism, $D$ = angle of minimum deviation, $\mu$ = refractive index.

Dispersion of White Light

White light is dispersed into its component colours (spectrum) because the refractive index of glass is different for different wavelengths. Red light deviates least; violet deviates most. This is why a prism produces a rainbow.

⚡ NECO Common Mistakes:

• Drawing angles from the surface instead of from the normal
• Getting confused about which direction light bends (towards normal when slowing down, away when speeding up)
• Forgetting that for total internal reflection, light must go from denser → rarer medium
• Mixing up $D$ (angle of deviation) and $i$ (angle of incidence) in prism questions

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for NECO and JAMB Physics preparation.

Derivation of Snell’s Law

Snell’s Law can be derived from Huygens’ wave theory. When light enters a different medium at angle $i$, the wavelength changes but frequency remains constant. Since $v = f\lambda$, if $v$ decreases, $\lambda$ decreases proportionally.

Consider wavefronts hitting a boundary: the phase change on refraction leads to: $$\frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$$

Critical Angle Derivation

When angle of refraction $r = 90°$, the angle of incidence is the critical angle $c$: $$\sin c = \frac{n_2}{n_1} \quad \Rightarrow \quad c = \sin^{-1}\left(\frac{n_2}{n_1}\right)$$

Lensmaker’s Equation: $$\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$ where $R_1$ and $R_2$ are radii of curvature of the two surfaces (positive if convex, negative if concave).

Applications of Refraction and Reflection

• Fibre optics: Total internal reflection allows light to travel through optical fibres with minimal loss. Medical endoscopes and telecommunications use this principle.
• Mirage: Refraction through layers of air at different temperatures creates fake images of distant objects.
• Rainbow: Dispersion + internal reflection in water droplets produces the rainbow.
• Eye glasses: Convex lenses correct farsightedness; concave lenses correct nearsightedness.

Two-Lens Systems

For a system of two lenses in contact: $\dfrac{1}{f} = \dfrac{1}{f_1} + \dfrac{1}{f_2}$ For separated lenses: use the image from the first lens as the object for the second.

Mirror and Lens Sign Conventions

Quantity	Real is Positive	Virtual is Negative
Object distance $u$	In front of mirror/lens	Behind
Image distance $v$	In front (real image)	Behind (virtual image)
Focal length $f$	Concave mirror / convex lens	Convex mirror / concave lens
Radius $R$	Centre of curvature in front	Behind surface

NECO/JAMB Question Patterns:

• NECO frequently asks: calculate critical angle given refractive index; determine image position using mirror/lens formula; find angle of deviation in a prism
• Watch for combined refraction problems (light going through multiple media)
• In practical questions, describe how to measure the focal length of a concave mirror using the distant object method

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