Combination of Thin LensesCombination of Thin Lenses | Equivalent Focal Length | Equivalent Power | Lens Systems | Numericals | PYQsIf any concept is not clear, contact Kumar Sir · +91-9958461445 · kumarsirphysics@gmail.comRay Optics · Complete Coaching Notes
Combination of Thin Lenses
Equivalent focal length, power, contact and separated systems, optical instruments, 30 solved numericals and a complete exam-practice bank.
1. Introduction and Applications
Multiple thin lenses are combined to obtain a focal length, power, field of view, magnification or aberration control that a single lens cannot provide conveniently.
TelescopeObjective + eyepiece
MicroscopeObjective + eyepiece
CameraMulti-element objective
SpectaclesVision correction
InstrumentsControl and correction
2. Two Thin Lenses in Contact
Let lens L₁ form an intermediate image at v₁. Because L₂ touches L₁, this intermediate image acts as the object for L₂ with object distance u₂=v₁.
For L₁: 1/f₁=1/v₁−1/u.
For L₂: 1/f₂=1/v−1/v₁.
Add both equations; the intermediate term cancels.
1/f₁+1/f₂=1/v−1/u.
For an equivalent lens, 1/F=1/v−1/u.
1/F = 1/f₁ + 1/f₂
Meaning: A contact combination behaves like one thin lens of focal length F.
3. Equivalent Power
P = 1/F(m)
Using 1/F=1/f₁+1/f₂ and Pᵢ=1/fᵢ:
P = P₁ + P₂
Positive power: converging contribution, usually convex in air.
Negative power: diverging contribution, usually concave in air.
+5 D and +2 D → +7 D.
+5 D and −2 D → +3 D.
4. Two Lenses Separated by Distance d
The first lens forms an intermediate image. For the second lens, the object distance is shifted by separation d. This translation produces the extra product term.
Use ray-transfer matrices: L(P)=[1 0; −P 1] and translation T(d)=[1 d; 0 1].
System matrix = L₂T(d)L₁.
Its power term is −[P₁+P₂−dP₁P₂].
Therefore P=P₁+P₂−dP₁P₂.
Since P=1/F, replace P₁=1/f₁ and P₂=1/f₂.
1/F = 1/f₁ + 1/f₂ − d/(f₁f₂)
d is the axial separation in the same length unit as f₁ and f₂. When d=0, the contact formula returns immediately.
Important: For separated lenses, F is the effective focal length. Front and back focal distances are measured from the system's principal planes, not automatically from either lens.
5. Special Cases
Combination
Power
Behavior
Key Result
Two convex lenses
Positive + positive
Stronger convergence
F smaller than either f in contact
Two concave lenses
Negative + negative
Stronger divergence
F negative with smaller magnitude
Convex + concave
Opposite signs
Depends on stronger lens
Sign of net power decides behavior
Equal and opposite focal lengths
Zero in contact
Afocal
F=∞
One lens stronger
Larger |P| dominates
Sign follows stronger power
Compare 1/|f|
6. Lens Systems and Practical Applications
Telescope: Long-focal-length objective forms a real intermediate image; eyepiece magnifies its angular size.
Compound microscope: Short-focal-length objective forms an enlarged intermediate image; eyepiece acts as a magnifier.
Camera: Several lens groups focus and correct aberrations while zooming changes effective focal length.
Human-eye correction: Spectacle/contact lens power combines with the eye's optical power to place the image on the retina.
Applications: zoom objectives, binoculars, projectors, medical endoscopes, corrective spectacles, photographic lenses and laboratory beam expanders.
7. Premium Formula Sheet
Contact: 1/F=1/f₁+1/f₂
Separated: 1/F=1/f₁+1/f₂−d/(f₁f₂)
Power: P=P₁+P₂
Separated power: P=P₁+P₂−dP₁P₂
Lens formula: 1/f=1/v−1/u
Magnification: m=v/u
8. Thirty Fully Solved Numerical Problems
1. Two convex lenses in contact
Question: f₁=20 cm and f₂=30 cm. Find F and P.
Given: f₁=0.20 m, f₂=0.30 m.
Formula: 1/F=1/f₁+1/f₂.
Substitution: P=5+3.333.
Calculation: P=8.333 D; F=0.12 m.
F=12 cm; P=+8.33 D.
Tip: Add powers in contact.
Mistake: Adding focal lengths directly.
2. Convex and concave in contact
Question: f₁=+25 cm, f₂=−50 cm. Find F.
Given: P₁=+4 D, P₂=−2 D.
Formula: P=P₁+P₂.
Substitution: P=4−2.
Calculation: P=+2 D.
F=+50 cm.
Tip: Sign decides convergence.
Mistake: Treating concave f as positive.
3. Two concave lenses
Question: f₁=−20 cm and f₂=−30 cm. Find F.
Given: f₁=−0.20 m, f₂=−0.30 m.
Formula: P=P₁+P₂.
Substitution: P=−5−3.333.
Calculation: F=1/(−8.333).
F=−12 cm.
Tip: Both powers are negative.
Mistake: Reporting positive F.
4. Powers given directly
Question: +6 D and −2 D lenses are in contact. Find net power and F.
Given: P₁=6 D, P₂=−2 D.
Formula: P=P₁+P₂.
Substitution: P=6−2.
Calculation: F=1/4 m.
P=+4 D; F=+25 cm.
Tip: Dioptres add directly.
Mistake: Converting D to cm incorrectly.
5. Afocal contact pair
Question: f₁=+20 cm and f₂=−20 cm. Find F.
Given: P₁=+5 D, P₂=−5 D.
Formula: P=P₁+P₂.
Substitution: P=0.
Calculation: F=1/0.
F=∞; combination is afocal.
Tip: Equal opposite powers cancel.
Mistake: Writing F=0.
6. Three lenses in contact
Question: Powers +2 D, +3 D and −1 D are in contact. Find F.
Given: Pᵢ as stated.
Formula: P=ΣPᵢ.
Substitution: P=2+3−1.
Calculation: P=4 D.
F=25 cm.
Tip: Extend power addition to any number.
Mistake: Averaging the powers.
7. Find unknown lens
Question: A +5 D system contains a +8 D lens. Find the second power.
Given: P=5 D, P₁=8 D.
Formula: P₂=P−P₁.
Substitution: P₂=5−8.
Calculation: P₂=−3 D.
Second lens is −3 D; f₂=−33.3 cm.
Tip: Negative answer means concave.
Mistake: Ignoring system sign.
8. Equivalent lens forms image
Question: +4 D and +1 D lenses in contact view an object at 30 cm. Find v.
Given: F=20 cm, u=−30 cm.
Formula: 1/F=1/v−1/u.
Substitution: 1/20=1/v+1/30.
Calculation: 1/v=1/60.
v=+60 cm.
Tip: First replace by equivalent lens.
Mistake: Applying lens formula separately without need.
9. Magnification of contact pair
Question: For problem 8 find m.
Given: v=60 cm, u=−30 cm.
Formula: m=v/u.
Substitution: m=60/(−30).
Calculation: m=−2.
Image is inverted and twice enlarged.
Tip: Negative m means inverted.
Mistake: Dropping sign of u.
10. Separated convex lenses
Question: f₁=20 cm, f₂=30 cm, d=10 cm. Find F.
Given: Values in cm.
Formula: 1/F=1/f₁+1/f₂−d/(f₁f₂).
Substitution: 1/F=1/20+1/30−10/600.
Calculation: 1/F=1/15.
F=15 cm.
Tip: Keep all lengths in one unit.
Mistake: Omitting d term.
11. Separated powers
Question: P₁=5 D, P₂=4 D, d=0.10 m. Find P.
Given: d in metres.
Formula: P=P₁+P₂−dP₁P₂.
Substitution: P=5+4−0.1(20).
Calculation: P=7 D.
F=14.29 cm.
Tip: d must be metre with dioptres.
Mistake: Using d=10 in power formula.
12. Convex + concave separated
Question: f₁=+20 cm, f₂=−40 cm, d=10 cm. Find F.
Given: Signed focal lengths.
Formula: separated-lens formula.
Substitution: 1/F=1/20−1/40−10/(−800).
Calculation: 1/F=3/80.
F=+26.67 cm.
Tip: Product f₁f₂ is negative.
Mistake: Losing the double negative.
13. Two concave lenses separated
Question: f₁=−20 cm, f₂=−30 cm, d=10 cm. Find F.
Given: Both f negative.
Formula: separated formula.
Substitution: 1/F=−1/20−1/30−10/600.
Calculation: 1/F=−1/10.
F=−10 cm.
Tip: Separation strengthens this divergence.
Mistake: Assuming d term always raises F.
14. Separation for zero power
Question: Two convex lenses f₁=20 cm, f₂=30 cm. Find d for afocal system.
Given: 1/F=0.
Formula: 0=1/f₁+1/f₂−d/(f₁f₂).
Substitution: d=f₁+f₂.
Calculation: d=20+30.
d=50 cm.
Tip: Keplerian telescope separation is f₁+f₂.
Mistake: Setting d=0 for afocal.
15. Unknown separation
Question: f₁=f₂=20 cm and F=25 cm. Find d.
Given: 1/25=1/20+1/20−d/400.
Formula: separated formula.
Substitution: d/400=0.10−0.04.
Calculation: d=24 cm.
Separation d=24 cm.
Tip: Solve algebra before substituting units.
Mistake: Using F=f₁+f₂.
16. Unknown second focal length
Question: Contact pair has F=15 cm and f₁=30 cm. Find f₂.
Given: 1/15=1/30+1/f₂.
Formula: contact formula.
Substitution: 1/f₂=1/30.
Calculation: f₂=30 cm.
f₂=+30 cm.
Tip: Compare powers quickly.
Mistake: Subtracting focal lengths.
17. Spectacle combination
Question: A +2.5 D correction is combined with −0.5 D coating equivalent. Find net f.
Given: P=2.5−0.5.
Formula: P=ΣP.
Substitution: P=2 D.
Calculation: f=0.5 m.
Net focal length +50 cm.
Tip: Add signed powers.
Mistake: Averaging prescriptions.
18. Camera lens group
Question: +10 D and +5 D groups are separated by 2 cm. Find system power.
Given: d=0.02 m.
Formula: P=P₁+P₂−dP₁P₂.
Substitution: P=10+5−0.02(50).
Calculation: P=14 D.
F=7.14 cm.
Tip: Short separation still matters at high power.
Mistake: Treating separated groups as contact.
19. Telescope objective and eyepiece
Question: fₒ=100 cm and fₑ=5 cm. Find normal-adjustment separation and angular magnification magnitude.
Given: Kepler telescope.
Formula: L=fₒ+fₑ; |M|=fₒ/fₑ.
Substitution: L=105; |M|=100/5.
Calculation: |M|=20.
Length=105 cm; magnification 20×.
Tip: Telescope formula is an afocal special case.
Mistake: Using contact formula.
20. Galilean telescope
Question: fₒ=80 cm, fₑ=−5 cm. Find normal-adjustment length and |M|.
Given: Concave eyepiece.
Formula: L=fₒ−|fₑ|; |M|=fₒ/|fₑ|.
Substitution: L=75; |M|=16.
Calculation: Direct.
Length=75 cm; magnification 16×.
Tip: Galilean telescope is shorter.
Mistake: Adding |fₑ|.
21. Microscope objective image
Question: Objective f=2 cm, object at 2.5 cm. Find v.
Given: f=2, u=−2.5 cm.
Formula: 1/f=1/v−1/u.
Substitution: 1/2=1/v+1/2.5.
Calculation: 1/v=0.1.
v=10 cm.
Tip: Objective makes real enlarged image.
Mistake: Taking u positive.
22. Compound system final image
Question: L₁ f=10 cm forms I₁ at 30 cm; L₂ is 20 cm right of L₁ with f₂=15 cm. Find final v₂.
Given: I₁ lies 10 cm right of L₂, so u₂=+10 cm (virtual object).
Formula: 1/f₂=1/v₂−1/u₂.
Substitution: 1/15=1/v₂−1/10.
Calculation: 1/v₂=1/6.
Final image 6 cm right of L₂.
Tip: Intermediate image may be a virtual object.
Mistake: Automatically making u₂ negative.
23. Sequential magnification
Question: If m₁=−2 and m₂=+3, find total magnification.
Given: Two-stage system.
Formula: m=m₁m₂.
Substitution: m=(−2)(3).
Calculation: m=−6.
Final image inverted; 6× size.
Tip: Magnifications multiply.
Mistake: Adding magnifications.
24. Net power in water
Question: Two lens powers in water are +2 D and −0.5 D. Find F.
Given: Powers already relative to water.
Formula: P=ΣP.
Substitution: P=1.5 D.
Calculation: F=1/1.5 m.
F=66.7 cm in water.
Tip: Do not reconvert stated medium powers.
Mistake: Using air values.
25. JEE-style separated system
Question: P₁=+10 D, P₂=−5 D, d=0.04 m. Find P.
Given: Signed powers.
Formula: P=P₁+P₂−dP₁P₂.
Substitution: P=10−5−0.04(−50).
Calculation: P=7 D.
F=14.29 cm.
Tip: Opposite powers make product negative.
Mistake: Missing plus from double negative.
26. AP/IB ray-system power
Question: A system has effective focal length 0.40 m. Find power.
Given: F=0.40 m.
Formula: P=1/F.
Substitution: P=1/0.4.
Calculation: P=2.5 D.
Equivalent power +2.5 D.
Tip: F sign determines P sign.
Mistake: Using centimetres in P=1/F.
27. IGCSE contact lenses
Question: Two converging lenses have f=40 cm each. Find contact F.
Given: Equal focal lengths.
Formula: 1/F=2/f.
Substitution: F=f/2.
Calculation: F=20 cm.
Equivalent focal length 20 cm.
Tip: Equal positive lenses halve f.
Mistake: Doubling f.
28. Image by equivalent concave pair
Question: Two −4 D lenses in contact view object at 25 cm. Find v.
Given: P=−8 D, F=−12.5 cm, u=−25 cm.
Formula: 1/F=1/v−1/u.
Substitution: −1/12.5=1/v+1/25.
Calculation: 1/v=−0.12.
v=−8.33 cm; virtual erect image.
Tip: Concave combination remains diverging.
Mistake: Making v positive.
29. Lens replacement
Question: Replace +3 D and +2 D contact lenses by one lens. Specify it.
Given: P=5 D.
Formula: f=1/P.
Substitution: f=0.2 m.
Calculation: Positive power.
Use one +5 D convex lens, f=20 cm.
Tip: Replacement matches equivalent power.
Mistake: Matching only one component.
30. Separation changes power
Question: Two +5 D lenses move from contact to 5 cm separation. Find power change.
Given: d=0.05 m.
Formula: Psep=10−d(25).
Substitution: Psep=10−1.25.
Calculation: Psep=8.75 D.
Power falls from 10 D to 8.75 D; F=11.43 cm.
Tip: Positive-positive separation reduces net power.
Mistake: Assuming power unchanged.
9. Important Previous Year Questions
Exam-style questions based on previous exam patterns; exact years are not claimed.
CBSE / NEET
Derive the equivalent focal length of two lenses in contact.
Apply the lens formula to each lens; the intermediate-image term cancels, yielding 1/F=1/f₁+1/f₂.
A +4 D and −1 D pair is in contact. Find F.
P=3 D, so F=1/3 m=33.3 cm.
Why are multiple elements used in a camera objective?
To obtain desired focal length/zoom and reduce chromatic, spherical and geometric aberrations.
JEE Main / Advanced
Derive separated-lens power using matrices.
Multiplying L₂T(d)L₁ gives system C=−(P₁+P₂−dP₁P₂), hence P=P₁+P₂−dP₁P₂.
Find the afocal separation of two positive lenses.
Set P=0: d=(P₁+P₂)/(P₁P₂)=f₁+f₂.
Can equivalent focal length alone locate focal points from each lens?
Not generally. A separated system has shifted principal planes, so front/back focal distances require principal-plane information.
IGCSE / A-Level / IB / AP
Explain how two converging lenses affect system power.
In contact their positive powers add. With separation d, the product term reduces the net power.
Design a method to measure equivalent focal length.
Focus a distant object sharply on a screen; measure from the appropriate principal plane. For a compact contact pair, the pair's central plane is a useful approximation.
Explain the role of objective and eyepiece in a telescope.
The objective forms a real intermediate image; the eyepiece magnifies its angular size and, at normal adjustment, sends parallel rays to the eye.
10. Twenty-Five MCQs
MCQs 1–13
1. +3 D and +2 D in contact give: A 1 D B 5 D C 6 D D −1 D
B. Powers add.
2. +5 D and −5 D in contact are: A converging B diverging C afocal D mirror
C. Net power zero.
3. Contact F for 20 cm and 30 cm is: A 50 B 25 C 12 D 10 cm
C. F=f₁f₂/(f₁+f₂)=12 cm.
4. SI unit of power: A metre B dioptre C watt D radian
B.
5. For separated lenses the extra term is: A +d/f₁f₂ B −d/f₁f₂ C d²/f₁f₂ D zero
B.
6. If d=0, separated formula becomes: A mirror formula B contact formula C Newton formula D Snell law
B.
7. Two concave lenses have net power: A positive B negative C zero always D undefined
B.
8. Stronger lens means: A larger |f| B smaller |P| C larger |P| D zero P
C.
9. Powers +8 D and −3 D give F: A 20 cm B 5 cm C −20 cm D 11 cm
A. P=5 D.
10. Magnifications of stages combine by: A addition B multiplication C subtraction D averaging
B.
11. Two +5 D lenses separated 0.1 m have P: A 10 B 7.5 C 5 D 12.5 D
B. 10−0.1×25=7.5 D.
12. Afocal positive pair separation is: A f₁−f₂ B f₁+f₂ C f₁f₂ D zero
B.
13. In P=P₁+P₂−dP₁P₂, d is in: A cm always B metre if P in D C degree D no unit
B.
MCQs 14–25
14. +2 D and −5 D contact pair is: A converging B diverging C afocal D plane
B. Net −3 D.
15. Effective F of P=−4 D: A +25 B −25 C −40 D +40 cm
B.
16. Telescope objective usually has: A short f B long f C negative f always D zero power
B.
17. Microscope objective usually has: A short positive f B long negative f C zero f D plane surfaces
A.
18. Intermediate image for lens 2 can be: A only real object B virtual object also C never object D mirror
B.
19. For contact lenses, order of lenses: A changes F B does not change F C reverses sign D doubles P
B.
20. For separated lenses, order can change: A equivalent power B principal-plane locations C P sign always D d
B. Effective power is symmetric, principal planes are not.
21. Three contact powers add as: A product B algebraic sum C average D reciprocal sum
B.
22. F=∞ means P: A ∞ B 1 C 0 D −∞
C.
23. A positive-negative pair is converging if: A |P+|>|P−| B opposite C equal D d=0 only
A.
24. Camera zoom changes primarily: A effective focal length B light speed C Planck constant D mirror radius
A.
25. Lens formula for equivalent contact lens is: A 1/F=1/v−1/u B 1/F=1/u+1/v C F=u+v D uv=F
