1cLenses and Image Formation
Apply the thin lens equation to converging and diverging lenses, classify images, and connect lens optics to cameras, eyeglasses, and microscopes.
Lenses are the heart of cameras, microscopes, telescopes, and corrective eyewear. Mastering the thin lens equation lets you design and analyze any optical instrument.
Lesson Overview
Lenses use refraction to converge or diverge light and form images. In this lesson you will distinguish converging and diverging lenses, apply the thin lens equation and magnification formula, classify images as real or virtual, and connect lens optics to everyday devices like cameras, eyeglasses, microscopes, and telescopes.
Key Concepts
Converging (Convex) Lens
Thicker at center; bends parallel rays toward the focal point on the far side. Focal length f > 0.
Diverging (Concave) Lens
Thinner at center; bends parallel rays away from the focal point on the same side as the incoming light. Focal length f < 0.
Thin Lens Equation
1/f = 1/dₒ + 1/dᵢ — same form as the mirror equation; sign conventions differ.
Magnification
m = hᵢ/hₒ = −dᵢ/dₒ; positive m → upright image; negative m → inverted image.
Real vs Virtual Images
Real images (dᵢ > 0) form on the far side of the lens and can be projected. Virtual images (dᵢ < 0) form on the same side as the object.
Applications
Cameras (real, inverted image on sensor), eyeglasses (correcting focal length), microscopes and telescopes (multi-lens systems).
An object is 30 cm from a converging lens with f = 10 cm. Find the image distance and magnification.
An object is 8 cm from a converging lens with f = 12 cm (object inside focal point). Find dᵢ and describe the image.
A diverging lens has f = −15 cm. An object is 45 cm away. Find dᵢ and m.
A camera lens (f = 50 mm) focuses on an object 2 m away. Find the image distance.
An object 4 cm tall is placed 20 cm from a converging lens with f = 8 cm. Find the image height.
A converging lens has f = 20 cm. An object is 60 cm away. Find dᵢ and classify the image.
Hint: Use 1/dᵢ = 1/f − 1/dₒ. A positive dᵢ means a real image on the far side of the lens.
Why does a diverging lens always produce a virtual, upright, diminished image?
Hint: For a diverging lens f < 0. Substitute into the thin lens equation and show that dᵢ is always negative.
A person is nearsighted (can't see far objects clearly). Should their corrective lens be converging or diverging? Why?
Hint: Nearsighted eyes converge light too strongly. The corrective lens must spread light out before it enters the eye.
A magnifying glass (converging lens, f = 5 cm) is held 3 cm from a stamp. Find the magnification.
Hint: Calculate dᵢ first, then m = −dᵢ/dₒ. Expect a positive magnification greater than 1.
Two thin lenses are placed in contact. Lens 1 has f₁ = 10 cm and Lens 2 has f₂ = −30 cm. What is the effective focal length?
Hint: For lenses in contact: 1/f_eff = 1/f₁ + 1/f₂.
Key Vocabulary
Converging Lens
A convex lens that is thicker at the center; refracts parallel rays to meet at the focal point. Focal length is positive.
Example: A magnifying glass is a converging lens that enlarges objects placed inside its focal length.
Diverging Lens
A concave lens that is thinner at the center; refracts parallel rays so they appear to diverge from a virtual focal point. Focal length is negative.
Example: Eyeglasses for nearsighted people use diverging lenses to spread light before it enters the eye.
Thin Lens Equation
1/f = 1/dₒ + 1/dᵢ; relates focal length, object distance, and image distance for a thin lens.
Example: With f = 10 cm and dₒ = 30 cm, the equation gives dᵢ = 15 cm.
Magnification (m)
m = hᵢ/hₒ = −dᵢ/dₒ; the ratio of image height to object height. Positive = upright; negative = inverted.
Example: m = −2 means the image is twice as tall as the object and inverted.
Interactive Practice — 5 Questions
A converging lens with f = 15 cm has an object at dₒ = 45 cm. What is dᵢ?
Which type of lens always produces a virtual, upright, diminished image?
An object inside the focal length of a converging lens produces:
Magnification m = −3 means the image is:
A camera uses which type of lens to form an image on its sensor?
Independent Practice
An object is 50 cm from a converging lens with f = 20 cm. Find dᵢ, m, and describe the image fully.
A diverging lens has f = −10 cm. An object is 30 cm away. Calculate dᵢ and m.
Explain why a converging lens acts as a magnifying glass only when the object is inside the focal point.
A farsighted person needs a converging lens with f = 40 cm. An object is at 25 cm. Where does the lens form the image? Is this useful for the eye?
★ A compound microscope has an objective lens (f = 0.5 cm) and an eyepiece (f = 2.5 cm). The object is 0.6 cm from the objective. Find the image distance from the objective, then explain qualitatively how the eyepiece further magnifies this image.
ChallengeCommon Mistakes
Using a positive focal length for a diverging lens.
Diverging (concave) lenses always have f < 0. Substitute the negative value into the thin lens equation.
Confusing the sign of dᵢ: assuming negative means behind the observer.
For lenses, positive dᵢ means the image is on the far side (real); negative dᵢ means on the same side as the object (virtual).
Forgetting to include the negative sign in m = −dᵢ/dₒ.
The negative sign is essential — it tells you whether the image is upright or inverted.
Math Tips
The thin lens equation is identical in form to the mirror equation. The key difference is the sign convention for dᵢ: for lenses, positive dᵢ is on the transmission side (far side); for mirrors, positive dᵢ is on the reflection side (front).
To find image height: first find m, then hᵢ = m × hₒ. A negative hᵢ simply means the image is inverted — the magnitude gives the actual size.