Reflection, refraction & lenses

Learning goals

What students should be able to explain, calculate and investigate after completing Reflection, Refraction and Lenses.

• State the laws of reflection and draw ray diagrams with angles measured from the normal.
• State Snell's Law and calculate the refractive index of a material.
• Define critical angle and total internal reflection, and state the condition for each to occur.
• Draw correct ray diagrams for convex and concave lenses, for different object positions.
• Use the lens formula to solve numerical problems, and describe image nature (real/virtual, upright/inverted, magnified/diminished) from a ray diagram.
• Describe the eye as an optical system and explain myopia and hypermetropia and their correction.

Key definitions

Precise terms, symbols and SI units used in Reflection, Refraction and Lenses.

Reflection

The law of reflection states that the angle of incidence equals the angle of reflection, both measured from the normal (an imaginary line perpendicular to the surface at the point of incidence).

The incident ray, reflected ray and normal lie in the same plane. Angles are measured from the normal, not from the mirror surface. A plane mirror forms a virtual image because reflected rays diverge but appear to originate behind the mirror. The image is upright, laterally inverted, the same size as the object and equally far behind the mirror.

Refraction and Snell's Law

Light changes speed (and usually direction) when it passes from one medium into another.

Refraction happens because wave speed changes at a boundary. One side of an oblique wavefront enters the new medium first and changes speed first. The wavefront pivots, so the ray—drawn perpendicular to the wavefront—changes direction. Frequency is fixed by the source and does not change; speed and wavelength change.

• n is the refractive index of the material relative to air.
• i is the angle of incidence, measured from the normal.
• r is the angle of refraction, measured from the normal.
• c₁ and c₂ are the speeds of light in the first and second media.
• In the general form, n₁ and n₂ are the refractive indices of the first and second media.

Critical Angle and Total Internal Reflection

At the critical angle, light travelling from a higher to a lower refractive index is refracted along the boundary; at larger incident angles it undergoes total internal reflection.

Total internal reflection can occur only when light travels from a higher refractive index to a lower refractive index. At the critical angle, the refracted ray travels along the boundary. For a larger incident angle there is no transmitted ray in the simple ray model: the light is reflected back into the denser medium.

Lenses

A convex (converging) lens brings parallel rays together at the principal focus; the type and position of the image it forms depends on where the object is placed relative to F and 2F.

A converging lens refracts rays that are initially parallel to the principal axis through the far principal focus. A ray through the optical centre is approximately undeviated, and a ray directed through the near focus emerges parallel to the axis. A diverging lens makes an incident ray parallel to the axis diverge as if it came from the near principal focus; a central ray remains approximately undeviated. Real images occur where rays actually meet; virtual images are located where backward extensions of diverging rays meet.

• For a real object, take u as positive.
• For a converging lens, f is positive; for a diverging lens, f is negative.
• A real image has positive v; a virtual image has negative v.
• Substitute the signs before rearranging the lens formula, then interpret the sign of the answer physically.

• A ray parallel to the principal axis emerges diverging as if it came from the near focus.
• A ray through the optical centre continues approximately straight.
• Backward extensions of the refracted rays locate the virtual, upright, diminished image.

Optional: The Eye

The eye focuses light using the cornea and the lens onto the retina, which acts like a screen.

The cornea does most of the refraction; the lens fine-tunes the focus by changing shape. In myopia, the eye focuses too strongly (or the eyeball is too long), so distant objects focus in front of the retina; a concave lens diverges the light slightly before it enters the eye, moving the focus back onto the retina. In hypermetropia, the eye does not focus strongly enough (or the eyeball is too short), so near objects focus behind the retina; a convex lens converges the light before it enters the eye, moving the focus forward onto the retina.

Diagrams and formulae

Key diagrams and equations for reviewing Reflection, Refraction and Lenses.

• Reflection: both the angle of incidence i and angle of reflection r are measured from the normal, not from the mirror surface.
• Refraction: light bends towards the normal when entering an optically denser medium, and away from the normal when leaving it.
• Total internal reflection: as the angle of incidence increases past the critical angle C, the ray changes from partially refracted (i < C), to refracted along the surface (i = C), to totally internally reflected (i > C).
• Convex lens: object placed at 2F produces a real, inverted image of the same size, also at 2F on the other side. The image nature for other object positions is summarised below.

Reflection: both the angle of incidence i and angle of reflection r are measured from the normal, not from the mirror surface.

Refraction: light bends towards the normal when entering an optically denser medium, and away from the normal when leaving it.

Total internal reflection: as the angle of incidence increases past the critical angle C, the ray changes from partially refracted (i < C), to refracted along the surface (i = C), to totally internally reflected (i > C).

Convex lens: object placed at 2F produces a real, inverted image of the same size, also at 2F on the other side. The image nature for other object positions is summarised below.

Concave lens: always produces a virtual, upright, diminished image located between the object and the lens, regardless of object position.

Eye defects: myopia focuses the image in front of the retina (corrected with a concave lens); hypermetropia focuses the image behind the retina (corrected with a convex lens).

Note: lens magnification (m = v/u) and lens power (P = 1/f) are in the old 2009 Tables book but are not part of the new Leaving Certificate Physics specification, so they have been left out of these notes. Image nature (real/virtual, upright/inverted, magnified/diminished) is determined from the ray diagram instead.

Worked Example

A step-by-step calculation applying the central relationships in Reflection, Refraction and Lenses.

A ray of light in air strikes the surface of a glass block at an angle of incidence of 40°. The refractive index of the glass is 1.5. Calculate the angle of refraction.

• n = sin i ÷ sin r
• 1.5 = sin 40° ÷ sin r
• sin r = sin 40° ÷ 1.5 = 0.643 ÷ 1.5 = 0.4287
• r = sin⁻¹(0.4287) ≈ 25.4°

Practical / Investigation

Experimental planning, measurement and evaluation for Reflection, Refraction and Lenses.

Experiment: Measuring the refractive index of glass using Snell's Law

• Apparatus: glass block, protractor, pins (or a ray box), paper.
• Method: trace the outline of the glass block; using pins (or a ray box), mark the path of a ray entering at a chosen angle of incidence, and its emerging path; join the points to find the refracted ray inside the block; measure i and r from the normal at the point of incidence.
• Repeat for several angles of incidence; plot sin i (y-axis) against sin r (x-axis) — the slope of the straight line gives the refractive index, n.
• Sources of error: inaccurate pin placement (parallax); the normal not drawn accurately perpendicular to the surface.
• Improvement: view pins at eye level to reduce parallax; use as large a glass block as practical to reduce the percentage error in measuring angles.

Experiment: Measuring critical angle and testing n = 1/sin C

• Use a semicircular glass or acrylic block and direct a narrow ray through the curved face towards the centre, so refraction at the entry face is minimised.
• Increase the internal angle of incidence at the flat face until the refracted ray just travels along the surface. Measure this angle from the normal as the critical angle C.
• Repeat and average C, then calculate n = 1/sin C and compare it with the refractive index obtained from the Snell’s-law graph. Avoid estimating C after total internal reflection has already begun.

Quick method — distant object: for a distant object, incoming rays are approximately parallel, so the image forms approximately at the focal plane. Focus a distant object (e.g. a tree or window seen through the far side of the room) on a screen and measure the distance from the lens to the screen. This gives an approximate focal length quickly, though it is less precise than the method below.

Experiment: Measuring the focal length of a converging lens (lens-formula method)

• Apparatus: converging lens, lens holder, illuminated object (or crosswire), screen, metre stick, optical bench (or bench of metre sticks).
• Method: place the object at a measured distance u from the lens; move the screen until a sharp, focused image forms, and measure the image distance v; repeat for several different values of u.
• For each pair of readings, calculate 1/u + 1/v; this should be constant (equal to 1/f) within experimental error, verifying the lens formula and giving a value for f.
• Sources of error: difficulty judging the sharpest focus by eye; measuring distances from the wrong reference point on the lens.
• Improvement: take repeat readings of v for each u and average; always measure distances from the centre of the lens.

• Refraction: measure i and r from the normal, repeat for several incident angles and plot sin i against sin r; gradient gives n for air-to-glass.
• Lens: measure u and v from the optical centre for several sharp images and use 1/f = 1/u + 1/v; repeat the screen position when judging focus.
• Align ray box/glass block or object/lens/screen on one axis. Never measure an angle from the surface instead of the normal.

• Use supplied object and virtual-image distances for a diverging lens, applying the sign convention stated with the dataset.
• Calculate 1/u + 1/v for several pairs and test whether it gives a consistent negative value of 1/f.
• Compare the calculated focal lengths, quote their spread or uncertainty and explain why a diverging lens cannot form a real image of a real object on a screen by itself.

Quick-Check Questions

Short retrieval questions covering the essential ideas in Reflection, Refraction and Lenses.

1. State the law of reflection.

2. Define critical angle.

3. What type of lens is used to correct myopia, and why?

4. What two rays are used to construct a lens ray diagram?

5. Why does light bend away from the normal when leaving a denser medium?

Common mistakes and exam tips

Common misconceptions and exam-technique checks for Reflection, Refraction and Lenses.

• Avoid: Measuring angles from the reflecting or refracting surface. Correct idea: Measure angles of incidence, reflection and refraction from the normal.
• Avoid: Using n = sin i/sin r without checking the direction of travel. Correct idea: This form gives the refractive index for light travelling from air or vacuum into the medium; define the two media for other cases.
• Avoid: Applying n = 1/sin C when light travels towards the denser medium. Correct idea: The relation applies for the critical angle when light travels from the medium into air or vacuum.

• Draw and label the normal before marking optical angles.
• State the direction of travel and the two media before applying a refractive-index relation.
• Use standard construction rays to determine the nature of a lens image.

Exam-style questions

Five exam-style questions with worked answers on Reflection, Refraction and Lenses.

Higher Level — Required Content

Required Higher Level content for this topic.

c₁ and c₂ are the speeds of light in the first and second media.

• At the critical angle, the refracted ray travels along the boundary.
• Total internal reflection occurs only from higher to lower refractive index and for incidence angles greater than C.

End-of-unit revision

Formulae, definitions, experiments and mixed revision questions for Reflection, Refraction and Lenses.

• Angle of incidence = angle of reflection (from the normal)
• n = sin i ÷ sin r (air/vacuum into medium)
• n = 1 ÷ sin C (medium into air/vacuum)
• 1/f = 1/u + 1/v (lens formula)

• Law of reflection
• Refractive index
• Critical angle and total internal reflection
• Principal focus and focal length
• Myopia and hypermetropia

• Reflection with angles from the normal
• Refraction bending toward/away from the normal
• The three TIR cases: i < C, i = C, i > C
• Convex lens ray diagram (object at 2F)
• Concave lens ray diagram (always virtual/upright/diminished)
• Myopia and hypermetropia with correcting lenses

• Measuring the refractive index of glass using Snell's Law (pins or ray box)
• Measuring the focal length of a converging lens

• Measuring angles from the surface instead of the normal
• Applying n = sin i/sin r or n = 1/sin C in the wrong direction
• Trying to get image orientation from a formula instead of the ray diagram
• Believing a concave lens can ever form a real image

1. Light travels from water (n = 1.33) into air. Calculate the critical angle for water.

2. A convex lens has a focal length of 0.2 m. An object is placed 0.5 m from the lens. Use the lens formula to find the image distance.

3. Explain why a fish looking up towards the water surface, at a viewing angle far from straight up, sees the surface act like a mirror rather than seeing out of the water.

4. Explain, in terms of image formation, why a person with hypermetropia struggles to focus on near objects without corrective lenses.

1. n = 1/sin C → sin C = 1/1.33 = 0.752 → C = sin⁻¹(0.752) ≈ 48.8°.

2. 1/f = 1/u + 1/v → 1/0.2 = 1/0.5 + 1/v → 5 = 2 + 1/v → 1/v = 3 → v ≈ 0.33 m.

3. When the angle of incidence, measured from the normal to the water surface, is greater than the critical angle, total internal reflection occurs and no light escapes — the surface instead reflects the underwater scene like a mirror.

4. In hypermetropia, the eye does not converge light strongly enough for near objects, so the image forms behind the retina, appearing blurred; a convex corrective lens adds extra convergence so the image forms correctly on the retina.

The table below shows which equations from this unit appear in the official State Examinations Commission Formulae and Tables book, and where.

Note: the new Leaving Certificate Physics specification's own equation list confirms the lens formula and refractive index — but does not include magnification or lens power, so image size/orientation is determined from the ray diagram rather than by calculation in these notes.