Physics Optics and Modern - How to Solve: Lens Formula (Concave/Convex, Lens Maker, Displacement Method) – IIT JEE Guide

How to Solve: Lens Formula (Concave/Convex, Lens Maker, Displacement Method) – IIT JEE Guide

Score Impact: 4–6 marks in JEE Main, 8–12 marks in JEE Advanced (direct + application in optics problems).

Introduction

"Master the lens formula, and you’ll solve 90% of optics problems in JEE—from microscope design to correcting vision defects. One formula, three methods, and zero guesswork."

WHAT YOU NEED TO KNOW FIRST

1. Sign Convention (Cartesian): Distances measured from the optical center; light travels left to right.
2. Real vs. Virtual Images: Real images form on the opposite side of the lens; virtual on the same side.
3. Focal Length Basics: Convex lenses have +f, concave have –f.

KEY TERMS & FORMULAS

1. Lens Formula

Formula: [ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} ] Variables: - ( v ) = Image distance (from optical center) - ( u ) = Object distance (from optical center) - ( f ) = Focal length MEMORISE THIS: Signs matter! Use Cartesian convention.

2. Lens Maker’s Formula

Formula: [ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) ] Variables: - ( n ) = Refractive index of lens material - ( R_1, R_2 ) = Radii of curvature (signs depend on direction) MEMORISE THIS: For plano-convex lenses, one ( R ) is infinite (e.g., ( R_2 = \infty )).

3. Displacement Method (for Focal Length)

Formula: [ f = \frac{D^2 - d^2}{4D} ] Variables: - ( D ) = Distance between object and screen - ( d ) = Distance between two lens positions Given on exam sheet: But memorize the derivation steps.

STEP-BY-STEP METHOD

Step 1: Identify the Lens Type

• Convex (Converging): +f, real focus.
• Concave (Diverging): –f, virtual focus.

Step 2: Assign Signs to Given Quantities

• Object distance ( u ): Always negative (light travels left to right).
• Image distance ( v ): Positive if real, negative if virtual.
• Focal length ( f ): Positive for convex, negative for concave.

Step 3: Plug into the Lens Formula

[ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} ] Solve for the unknown (( v ), ( u ), or ( f )).

Step 4: Check for Magnification (if needed)

[ m = \frac{v}{u} = \frac{\text{Image height}}{\text{Object height}} ] - ( m > 0 ): Virtual, erect image. - ( m < 0 ): Real, inverted image.

Step 5: Verify Units and Signs

• All distances in meters (or cm, but consistent).
• Double-check signs before final answer.

WORKED EXAMPLES

Example 1 – Basic (Convex Lens)

Problem: An object is placed 20 cm from a convex lens of focal length 10 cm. Find the image distance.

Solution: 1. Lens type: Convex → ( f = +10 ) cm. 2. Object distance: ( u = -20 ) cm (negative by convention). 3. Lens formula:
[ \frac{1}{v} - \frac{1}{-20} = \frac{1}{10} ]
[ \frac{1}{v} + \frac{1}{20} = \frac{1}{10} ] 4. Solve for ( v ):
[ \frac{1}{v} = \frac{1}{10} - \frac{1}{20} = \frac{1}{20} ]
[ v = +20 \text{ cm} ] 5. Interpretation: Positive ( v ) → Real image, 20 cm from lens.

What we did and why: - Used sign convention strictly. - Solved step-by-step to avoid arithmetic errors.

Example 2 – Medium (Lens Maker’s Formula)

Problem: A plano-convex lens has a radius of curvature of 30 cm and refractive index 1.5. Find its focal length.

Solution: 1. Lens type: Plano-convex → ( R_1 = +30 ) cm, ( R_2 = \infty ). 2. Lens Maker’s formula:
[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{30} - \frac{1}{\infty} \right) ]
[ \frac{1}{f} = 0.5 \times \frac{1}{30} = \frac{1}{60} ] 3. Focal length: ( f = +60 ) cm.

What we did and why: - Recognized ( R_2 = \infty ) for plano-convex. - Applied the formula directly with correct signs.

Example 3 – Exam-Style (Displacement Method)

Problem: A lens forms a sharp image on a screen when placed at positions 40 cm apart. The distance between object and screen is 100 cm. Find the focal length.

Solution: 1. Given: ( D = 100 ) cm, ( d = 40 ) cm. 2. Displacement formula:
[ f = \frac{D^2 - d^2}{4D} ]
[ f = \frac{100^2 - 40^2}{4 \times 100} = \frac{10000 - 1600}{400} = \frac{8400}{400} = 21 \text{ cm} ] 3. Interpretation: Positive ( f ) → Convex lens.

What we did and why: - Identified the method from the problem statement. - Plugged values directly into the formula.

COMMON MISTAKES

1. Mistake: Ignoring sign convention.
   Why it happens: Students forget ( u ) is always negative.
   Correct approach: Always write ( u = -|u| ).

2. Mistake: Confusing ( R_1 ) and ( R_2 ) in Lens Maker’s formula.
   Why it happens: Not visualizing lens shape.
   Correct approach: Draw the lens; ( R_1 ) is the first surface light hits.

3. Mistake: Misapplying displacement method.
   Why it happens: Not recognizing the two-lens-positions scenario.
   Correct approach: Only use this when the problem mentions two sharp images.

4. Mistake: Forgetting units.
   Why it happens: Carelessness.
   Correct approach: Always convert to meters (or cm consistently).

5. Mistake: Assuming all images are real.
   Why it happens: Overlooking concave lenses.
   Correct approach: Check ( f ) sign first.

EXAM TRAPS

1. Trap: Giving object distance as positive.
   How to spot it: Problem says "placed 20 cm from lens" without specifying direction.
   How to avoid it: Always assume ( u = -|u| ).

2. Trap: Mixing up ( R_1 ) and ( R_2 ) in Lens Maker’s formula.
   How to spot it: Problem describes a lens with one flat side.
   How to avoid it: Sketch the lens; ( R_2 = \infty ) for plano-convex.

3. Trap: Displacement method with incorrect ( D ) or ( d ).
   How to spot it: Problem mentions "two positions" but doesn’t specify distances.
   How to avoid it: Label ( D ) (object-screen) and ( d ) (lens shift) clearly.

1-MINUTE RECAP

"This is your last-minute lens formula cheat sheet. First, memorize the lens formula: ( 1/v - 1/u = 1/f ). Signs are everything: ( u ) is always negative, ( f ) is positive for convex, negative for concave. For Lens Maker’s, remember ( (n-1)(1/R_1 - 1/R_2) )—plano-convex means one ( R ) is infinite. Displacement method? ( f = (D^2 - d^2)/4D ). Double-check signs, units, and whether the image is real or virtual. Now go crush that optics problem!