How does power combine for lenses in contact?

For n lenses in contact, total power = sum of individual powers. P_eff = P1 + P2 + P3 + ... For 4 identical lenses each with power p: P_eff = p+p+p+p = 4p.

How does magnification combine for lenses in series?

For lenses in series, total magnification = product of individual magnifications. m_eff = m1 x m2 x m3 x ... For 4 identical lenses each with magnification m: m_eff = m x m x m x m = m^4.

What is the difference between power and magnification combination?

Power: ADDS (like resistances in parallel). For n identical lenses: P = np. Magnification: MULTIPLIES. For n identical lenses: M = m^n.

Why do magnifications multiply but powers add?

Powers add because focal lengths add in a specific way (1/f = 1/f1 + 1/f2 for contact). Magnifications multiply because each lens operates on the image formed by the previous one: final image size = object size x m1 x m2 x m3 x m4.

In a certain camera, a combination of four similar thin convex lenses are arranged axially in contact. Then the power of

Q: What is the unit of lens power?

Power P = 1/f where f is focal length in metres. Unit is Dioptre (D). Positive power = converging lens. Negative power = diverging lens.

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In a certain camera, a combination of four similar thin convex lenses are arranged axially in contact. Then the power of the combination and the total magnification in comparison to power p and magnification m for each lens will be respectively:

Options

p^4 and m^4

4p and 4m

p^4 and 4m

4p and m^4

Correct Answer

4p and m^4

Solution

Power of lenses in contact: P_eff = P1 + P2 + P3 + P4

All identical with power p: P_eff = 4p

Magnification of lenses in series: m_eff = m1 x m2 x m3 x m4

All identical with magnification m: m_eff = m x m x m x m = m^4

Answer: 4p and m^4

Power adds: 4p | Magnification multiplies: m^4
Remember: Power = sum, Magnification = product

Theory: Optics

1. Lenses in Contact

When lenses are in contact (touching): equivalent focal length 1/f = 1/f1 + 1/f2 + ... Equivalent power P = P1 + P2 + ... Powers simply add. For n identical lenses of power p each: P_total = np. Applications: camera lenses (multiple elements to correct aberrations), microscope objectives, eyepieces.

2. Magnification in Lens System

For a system of lenses in series (not in contact but in sequence): total magnification = product of individual magnifications. m_total = m1 x m2 x m3... This is because each lens takes the image of the previous as its object. For n identical lenses: m_total = m^n. Compare with power (sum) vs magnification (product).

3. Lens Formula and Magnification

Lens formula: 1/v - 1/u = 1/f. Magnification: m = v/u = (f)/(f+u) for thin lens. For convex lens: can give magnified, diminished, or equal image depending on object position. Inside focal length: virtual, erect, magnified (magnifying glass). At 2f: real, inverted, equal size. Between f and 2f: real, inverted, magnified. Beyond 2f: real, inverted, diminished.

4. Power of a Lens

Power P = 1/f (f in metres). Unit: dioptre (D). Converging lens: positive power. Diverging lens: negative power. Combination in contact: P = P1 + P2. Example: +3D and -1D lenses in contact give +2D = converging lens of focal length 50 cm. Spectacle lenses: prescription written in dioptres. Myopia: negative power (diverging). Hyperopia: positive power (converging).

5. Lens Aberrations

Chromatic aberration: different wavelengths refract differently, different focal lengths. Corrected by achromatic doublet (convex + concave lens of different glass). Spherical aberration: rays from edge of lens focused at different point than rays through centre. Corrected by aspheric lenses or lens combinations. Cameras use multiple lens elements (6-20) to correct various aberrations. Telescope objectives: large aperture requires careful aberration correction.

6. Human Eye and Accommodation

Human eye: biconvex crystalline lens of variable power (accommodation). Power range: 40-44 D (approximately). Far point: point on which unaccommodated eye focuses. Normal: infinity. Near point (least distance of distinct vision): 25 cm for normal adult eye. Accommodation: ciliary muscles change lens curvature (power). Presbyopia: loss of accommodation with age (need reading glasses). Power of eye system: cornea contributes ~43D, lens ~20D (variable).

7. Optical Instruments

Simple microscope (magnifying glass): single convex lens. M = 1 + D/f (D = 25 cm). Compound microscope: objective (short f, high power) + eyepiece. Total M = (L/fo)(1+D/fe) where L = tube length. Telescope (astronomical): objective (long f, large aperture) + eyepiece (short f). Magnification M = fo/fe. For viewing: final image at infinity, M = fo/fe. Resolving power of telescope: R = D/1.22lambda (larger aperture = better resolution). Camera: inverted real image on film/sensor. f-number = f/D (aperture).

8. Total Internal Reflection

When light travels from denser to rarer medium at angle greater than critical angle: total internal reflection occurs. Critical angle C: sin C = 1/n (n = refractive index of denser medium). Applications: optical fibres (light trapped by TIR, information transmitted over long distances with minimal loss), diamonds (cut to maximize TIR, giving brilliance), mirage (hot air near ground acts as rarer medium), prism binoculars (two TIR prisms fold optical path).

Frequently Asked Questions

1. Why does power add but magnification multiply? ⌄

Power P = 1/f. For lenses in contact: 1/f_eq = 1/f1 + 1/f2 + ... So P_eq = P1 + P2 + ... (addition). Magnification: each lens forms an image that becomes the object for the next lens. Final height = h x m1 x m2 x ... (multiplication). Fundamentally different physical processes: power relates to bending of light rays (additive), while magnification relates to scaling of image size (multiplicative).

2. What is the unit of lens power and what does dioptre mean? ⌄

Dioptre (D) = 1/metre. A lens of power 1D has focal length 1 metre = 100 cm. A lens of power 4D has focal length 25 cm. For the 4 lenses in this problem: if each has power p = 2D (f = 50 cm), combination has power 4p = 8D (f = 12.5 cm). The dioptre system makes it easy to calculate combinations: just add powers. This is why opticians use dioptres for prescriptions.

3. How does a compound microscope achieve high magnification? ⌄

Compound microscope magnification M = m_objective x m_eyepiece. Objective: short focal length (a few mm), placed just beyond focal point, creates large magnified intermediate image. M_obj = L/f_obj where L = tube length (distance from objective back focal point to eyepiece front focal point, ~15 cm). Eyepiece: acts as simple magnifier on intermediate image. M_eye = D/f_eye (D = 25 cm). Total M = (L/f_obj)(D/f_eye). For f_obj = 2 cm, f_eye = 5 cm: M = (15/2)(25/5) = 7.5 x 5 = 37.5x. Modern high-power microscopes can reach 1000x-2000x.

4. What happens when 4 identical diverging lenses are in contact? ⌄

For diverging lenses, power is negative (p < 0). Four identical diverging lenses of power p each: P_total = 4p (negative). The combination is also diverging with power |4p|. Magnification: m_total = m^4. Since m for each diverging lens is between 0 and 1 (virtual, diminished image), m^4 is even smaller (further diminished). This is why cameras avoid putting multiple diverging elements in contact.

5. How are lenses used in spectacle prescriptions? ⌄

Myopia (short-sightedness): eye is too powerful, images form in front of retina. Correction: diverging (concave) lens, negative power. Example: -2.5 D means focal length = -40 cm. Hyperopia (long-sightedness): eye is too weak, images form behind retina. Correction: converging (convex) lens, positive power. Presbyopia: loss of accommodation. Needs reading glasses (+1 to +3 D). Astigmatism: different power in different meridians. Correction: cylindrical or toric lens with different power in two perpendicular directions.

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