Question
A convex lens has a focal length of 10 cm. An object is placed 30 cm in front of it. Find the position, nature, and magnification of the image formed.
(Object is beyond 2F since 30 cm > 2 × 10 = 20 cm)
Solution — Step by Step
Set up the sign convention
We use the New Cartesian Sign Convention: distances measured in the direction of incident light are positive; opposite is negative.
- Object distance:
u = -30 cm(object is on the left) - Focal length:
f = +10 cm(convex lens, always positive)
Apply the Lens Formula
The lens formula is:
Substituting values:
Solve for v
Positive v means the image forms on the other side of the lens — the transmission side.
Find magnification
The negative sign tells us the image is inverted. The magnitude 0.5 tells us it is diminished (half the object size).
State the nature of the image
Image is: Real, Inverted, Diminished, formed at 15 cm on the other side of the lens (between F and 2F).
Why This Works
When an object is placed beyond 2F, the refracted rays converge on the other side — but closer to the lens than 2F. This is why our image lands at 15 cm (between F₂ and 2F₂), not beyond 2F₂.
The lens formula 1/v − 1/u = 1/f is derived from the geometry of similar triangles formed by the principal ray and the parallel ray through the lens. Every term carries sign information, which is why getting the signs right is non-negotiable.
This exact setup — object at 3F giving image at 1.5F — is a favourite for CBSE numericals because the numbers work out cleanly. In CBSE 2024 Board Exam, this appeared with these exact values.
Alternative Method — Ray Diagram Verification
We can verify without calculation using the two standard rays:
- Ray parallel to principal axis → passes through F₂ after refraction
- Ray through optical centre → passes straight without bending
These two rays meet at 15 cm on the other side. The ray diagram gives the same answer and is useful to sketch in board exams for the 1-mark "draw the diagram" part.
💡 Expert Tip
In board exams, always sketch a rough ray diagram even for numerical questions. It costs you 30 seconds and often fetches a separate mark for "diagram" in the marking scheme.
Common Mistake
⚠️ Common Mistake
The most common error is writing u = +30 instead of u = -30. Students forget that the object is on the incident side, so its distance is negative in Cartesian convention. With u = +30, you'd get 1/v = 1/10 − 1/30 = 2/30, which accidentally gives the same numerical answer — but for the wrong reasons, and this luck runs out in trickier problems. Always assign signs first, before plugging into the formula.
Also watch out for confusing the mirror formula (1/v + 1/u = 1/f) with the lens formula (1/v − 1/u = 1/f). The sign between the two terms on the left flips. This single character has cost students marks in board exams more times than we can count.