29Refraction through Lenses

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REFRACTION THROUGH LENSES
▢ Types of Lenses:

Table 1: Types of Lenses

Type
Examples
Double lens
Double convex lens, double concave lens
Plano lens
Plano-convex lens, plano-concave lens
Meniscus lens
Concavo-convex lens
❖ Main Classification:
  • •
    Convex lens → converging lens
  • •
    Concave lens → diverging lens
▢ Lens Maker's Formula:
❖ Formula: \(\frac{1}{f}=\left(\frac{\mu_g}{\mu_m}-1\right)\left(\frac{1}{R_1}+\frac{1}{R_2}\right)\)

Table 1: Symbols in Lens Maker's Formula

Symbol
Meaning
\(f\)
Focal length of lens
\(R_1\)
Radius of curvature of first curved surface
\(R_2\)
Radius of curvature of second curved surface
\(\mu_g\)
Refractive index of glass / lens material
\(\mu_m\)
Refractive index of surrounding medium
❖ Sign Convention in Note:
  • •
    For convex lens: \(f,R_1,R_2\) are positive
  • •
    For concave lens: \(f,R_1,R_2\) are negative
❖ Important Point: Focal length depends on refractive index of lens, refractive index of surrounding medium and radii of curvature
▢ Newton's Formula for Lens:
❖ Definition: Relation between focal length and distances of object and image from respective foci
❖ Formula: \(f=\sqrt{xy}\)
❖ Square Form: \(f^2=xy\)
❖ _*table:
    ❖ Different Media on Two Sides:
    ◉ Formula: \(f_1f_2=xy\)
    ◉ Meaning: \(f_1\) and \(f_2\) are focal lengths in two different media
    ▢ Thin Lenses Combination:
    ❖ **table:
      ❖ Power Form: \(P*{eq}=P_1+P_2\) for lenses in contact
      ▢ Displacement Method for Convex Lens:
      ❖ Purpose: Used to determine focal length of convex lens

      Table 1: Symbols in Displacement Method

      Symbol
      Meaning
      \(L_1\)
      Initial position of lens
      \(L_2\)
      Final position of lens
      \(D\)
      Distance between object and screen
      \(d\)
      Distance between two positions of lens
      \(f\)
      Focal length of convex lens
      ❖ Conditions:
      • •
        For convex lens, minimum distance between object and real image is \(4f\)
      • •
        For two sharp image positions, \(D>4f\)
      • •
        In two positions, object and image distances are interchanged
      • •
        \(u_1=v_2\)
      • •
        \(u_2=v_1\)
      ❖ Formulae:

      Table 1: Displacement Method Formulae

      Quantity
      Formula
      Focal length
      \(f=\frac{D^2-d^2}{4D}\)
      Object distance in first position
      \(u_1=\frac{D-d}{2}\)
      Image distance in second position
      \(v_2=\frac{D-d}{2}\)
      Object distance in second position
      \(u_2=\frac{D+d}{2}\)
      Image distance in first position
      \(v_1=\frac{D+d}{2}\)
      Magnification in first position
      \(m_1=\frac{D+d}{D-d}\)
      Magnification in second position
      \(m_2=\frac{D-d}{D+d}\)
      Magnification ratio
      \(\frac{m_1}{m_2}=\left(\frac{D+d}{D-d}\right)^2\)
      Product of magnifications
      \(m_1m_2=1\)
      Object-image size relation
      \(O^2=I_1I_2\)
      Magnification difference
      \(m_1-m_2=\frac{d}{f}\)
      Focal length using magnification
      \(f=\frac{d}{m_1-m_2}\)
      ❖ Magnification Nature:
      • •
        \(m_1>1\)
      • •
        \(m_2<1\)
      • •
        One image is magnified and the other is diminished
      ▢ Special Behaviour of Lenses:

      Table 1: Lens Behaviour in Different Conditions

      Condition
      Behaviour / Result
      Spherical air bubble in water
      Behaves as diverging lens / concave lens
      Spherical water droplet in air
      Behaves as convex lens
      Angle of incidence > critical angle for air bubble in water
      Air bubble behaves as convex mirror
      Half lens covered with black paper
      One complete image is formed but intensity is reduced
      Lens made of two different materials
      Two images of object are formed
      Convex lens in contact with mirror and space filled with water
      Power decreases
      Power of goggles spectacles
      Zero
      Power of viewing glass
      Infinity
      ▢ Minimum Object-Image Distance:

      Table 1: Minimum Distance for Real Image

      Optical System
      Minimum distance between object and real image
      Thin convex lens
      \(4f\)
      If distance < \(4f\)
      Image becomes virtual
      Concave mirror
      0
      ▢ Motion of Object and Image:
      ❖ _*table:
        ▢ Cutting and Displacing Convex Lens:
        ❖ Condition: Biconvex lens cut longitudinally along principal axis into two parts and parts displaced laterally
        ❖ Incident Light: Parallel beam of light
        ❖ Result: Two images are formed
        ❖ Intensity: Each image has reduced intensity
        ▢ Focal Length of Two Convex Lenses Separated by Distance:
        ❖ Point: Focal length of combination of two thin convex lenses separated by distance first increases and then decreases
        ❖ Formula: \(\frac{1}{f*{eq}}=\frac{1}{f_1}+\frac{1}{f_2}-\frac{x}{f_1f_2}\)
        ▢ Read and Digest:

        Table 1: Important Points

        Fact
        Answer
        Air bubble in water
        Diverging lens / concave lens
        Water droplet in air
        Convex lens
        Air bubble in water with \(i>C\)
        Convex mirror
        Half lens covered
        Image remains complete but less intense
        Minimum distance for real image by convex lens
        \(4f\)
        Object-image distance less than \(4f\)
        Virtual image
        Concave mirror minimum object-real image distance
        0
        Insect moving towards first focus of convex lens
        Image speed decreases
        Lens of two materials
        Two images form
        Two convex lenses separated by distance
        Equivalent focal length first increases then decreases
        Convex lens + mirror + water
        Power decreases
        Goggles spectacles power
        Zero
        Viewing glass power
        Infinity
        Biconvex lens cut longitudinally and displaced laterally
        Two images of reduced intensity
        ▢ High-Yield Recall:
        ❖ **table:
          Q1.
          The power of convex lens is P₁ = +6D and power of concave lens is P₂ = -4D. The focal length of combinations is:
          📅BP 2013
          Q2.
          If m = 2, R = 40 cm for a concave mirror. Then find the position of object i.e. object distance, u is:
          📅BP 2012
          Q3.
          Two thin lenses of focal length +60 cm and -20 cm are placed in contact. The focal length of combination is:
          📅BP 2011
          Q4.
          A body of size 1 m is on the axis of convex lens at its focus. Then size (height) of its image will be:
          📅BP 2010
          Q5.
          The power of combination of convex and concave lens is 4D. If the power of convex lens is 4D, the focal length of concave lens is:
          📅IOM 2014
          Q6.
          P and q are the distances of object and image from the principle focus of an equiconvex lens. Newton's formula for its focal length is:
          📅MOE 2012
          Q7.
          A biconvex lens of 8 cm and 12 cm radius of curvature with refractive index 1.5 has focal length:
          📅MOE 2011
          Q8.
          If a biconvex lens is silvered on one side, it will behave as:
          📅IOM 2013
          Q9.
          A convex lens has focal length 20 cm. Its power is:
          📅IOM 2012
          Q10.
          Two thin lenses of focal length f₁ and f₂ are placed at distance 'd'. For the power of combination to be zero, the separation 'd' is:
          📅IOM 2011
          Q11.
          Find the final image formed by lens combination (f₁ = 10 cm, f₂ = -5 cm, f₃ = 30 cm):
          📅IE 2013
          Q12.
          Two thin lenses (10 cm and 20 cm focal lengths) are placed in contact. Their combined focal length is:
          📅KU 2013, 2012
          Q13.
          If five lenses shown are made of the same material, which has the shortest positive focal length?
          📅KU 2011
          Q14.
          The ratio of powers of convex and concave lenses is 2/3 and their combined focal length is 30 cm. Individual focal lengths are:
          📅IE 2013
          Q15.
          A plano-convex lens has radius of curvature 10 cm and focal length 30 cm. Its refractive index is:
          📅MOE 2010
          Q16.
          A plano-convex lens is silvered at the plane surface. If radius of curvature is R and refractive index is n, the radius of curvature of the convex mirror formed is:
          📅IOM 07
          Q17.
          The distance between an object and a diverging lens is 'p' times the focal length. The lateral magnification 'm' is:
          📅IOM 06
          Q18.
          The focal length of a lens in air is 30 cm. In water (μw = 1.33, μlens = 1.5), its focal length is:
          📅IOM 02
          Q19.
          Two lenses of power +12D and -2D are placed in contact. The focal length of the combination is:
          📅MOE 06
          Q20.
          The effective power if lenses of focal length +10 cm and -20 cm are combined is:
          📅MOE 2056
          Q21.
          If an object is placed at the focus of a convex lens, the refracted rays are:
          📅MOE
          Q22.
          When a convex lens (f = 12 cm) is immersed in water, its focal length becomes:
          📅Bangladesh Emb
          Q23.
          An object is placed left of a convex lens forming an image on a screen. If the screen is shifted away:
          📅KU 08
          Q24.
          A convex lens (f = 0.5 m) and concave lens (f = 1 m) are combined. The power of the resulting lens is:
          📅KU 09, 2014
          Q25.
          A convex lens is dipped in a liquid with refractive index equal to the lens. Its focal length becomes:
          📅IE-04
          Q26.
          Image from a convex lens is formed beyond 1.5F. The object should be placed at:
          📅IE-05
          Q27.
          An object is placed 1 cm from a lens with magnification 5. Its focal length is:
          📅IE-06
          Q28.
          A converging lens forms an image 1.5f beyond the lens. The object is:
          📅TE-07
          Q29.
          An object is placed 10 cm in front of a diverging lens (f = -20 cm). The image will be:
          📅IE-01
          Q30.
          A plano-convex lens (μ = 1.5, R = 20 cm) has focal length:
          📅BPKIHS-08
          Q31.
          A plano-convex and plano-concave lens (radii R, refractive indices μ₁ and μ₂) have combined focal length:
          📅BPKIHS-09
          Q32.
          Two lenses (P = +1.75D and -1.25D) are combined. The focal length of the combination is:
          📅BPKIHS-97
          Q33.
          Two identical plano-convex lenses (f = 40 cm) are pressed together. To obtain a real, inverted image with magnification unity, the object distance is:
          📅BPKIHS
          Q34.
          The focal length of a convex lens is f. An object is placed at distance x from its first focal point. The ratio of image size to object size is:
          📅MOE
          Q35.
          A convex lens produces a real image n times the size of the object. The object distance is:
          📅MOE
          Q36.
          A convex lens produces a virtual image n times the size of the object. The object distance is:
          📅MOE
          Q37.
          A concave lens of focal length f produces an image 1/n times the size of the object. The object distance is:
          📅MOE
          Q38.
          A plano-convex lens (μ, radius R) is silvered on the plane side. The system behaves like a concave mirror of radius:
          📅MOE
          Q39.
          The distance between a convex lens and a plane mirror is 10 cm. Parallel rays incident on the lens form an image at the optical center after reflection. The focal length of the lens is:
          📅MOE
          Q40.
          A convex lens (f = 20 cm) and concave lens (f = -5 cm) are coaxial. A parallel beam leaves as a parallel beam. The separation between lenses is:
          📅MOE
          Q41.
          A lens (focal length f, aperture diameter d) forms an image of intensity I. If the central part (d/2 diameter) is blocked, the new intensity is:
          📅IOM 2017
          Q42.
          An object is placed 20 cm from a convex lens (f = 10 cm). The image is formed at:
          📅MOE
          Q43.
          Two thin lenses (f₁, f₂) are placed at distance 'd'. For zero power, the separation 'd' is:
          📅MOE
          Q44.
          A convex lens (f₁) and concave lens (f₂) in contact act as a convergent lens if:
          📅MOE
          Q45.
          A convex lens (+6D) and concave lens (-4D) in contact form a combination with:
          📅MOE
          Q46.
          A plano-convex lens (R = 10 cm, f = 30 cm) has refractive index:
          📅MOE
          Q47.
          A double convex lens (μ = 1.5, R = 20 cm) converges parallel rays at a distance:
          📅MOE
          Q48.
          A lens (f in air, μ = 1.5) is placed in liquid (μ = 1.33). Its focal length becomes:
          📅MOE
          Q49.
          For a convex lens (fv, fr) and concave lens (Fv, Fr):
          📅MOE
          Q50.
          An equiconvex lens (f = 0.1 m) is cut into two equal parts perpendicular to the axis. The ratio of new focal lengths is:
          📅MOE
          Q51.
          A symmetric double convex lens (P = 4D) is cut into two equal parts. The power of each part is:
          📅MOE
          Q52.
          The focal length of a plano-convex lens equals the radius of curvature of its curved surface. The refractive index is:
          📅MOE
          Q53.
          Rays from a luminous object focus at point A. A convex lens (f = 30 cm) is placed 30 cm from A. The new focus is at B. The distance AB is:
          📅MOE
          Q54.
          A convex lens forms a 4 cm image on a screen. When shifted, it forms a 16 cm image. The object length is:
          📅MOE
          Q55.
          A convex lens forms images with magnifications 2 and 0.5 for two positions separated by 30 cm. Its focal length is:
          📅MOE
          Q56.
          A lens forms a real image on a screen 100 cm from the object. When moved 20 cm, another image forms. The focal length is:
          📅MOE
          Q57.
          For a convex lens, maximum power occurs when:
          📅MOE
          Q58.
          An object (1.5 cm) is placed on the axis of a convex lens (f = 25 cm). A real image forms at 75 cm. The image height is:
          📅MOE
          Q59.
          A concavo-convex lens (R₁ = 40 cm, R₂ = 20 cm, μ = 1.5) has focal length:
          📅MOE
          Q60.
          An aeroplane with a camera (f = 5 cm) photographs 5 km terrain on 5 cm film. The flying height is:
          📅MOE
          Q61.
          An aeroplane is flying at a height of 1500m\n. It has a camera having convex lens of\nfocal length 45 cm with photographic plate\n30cm x 30cm. How much area on the\nground can be photographed at one time ?
          Q62.
          A cyclist is moving perpendicular to\nprincipal axis at a distance of 10m with a\nspeed of 10m/s infront of a convex lens of\nfocal length 10cm. Find the time of\nexposure of the lens if the image\ndisplaced by 1mm on the photographi\nplate.
          Q63.
          A plane convex lens has diameter 6cm and\nthickness from the centre is 3mm. If the\n\nspeed of light in the lens is 2x 10 m/s, then\nthe focal length of plane convex lens is
          Q64.
          A picture of size 2cm x 4em is shown on a\nprojector. If the magnification produced\nbe 10, the area of the image on the screen\nwill be
          Q65.
          If lens behaves as converging in air and\ndiverging in water. Then refractive index is\n[TOM 20151
          📅TOM 20151
          Q66.
          When the convex lens of refractive index\n(H), immersed in water of same refractive\nindex (1) then, its focal length is:
          📅KU 2016
          Q67.
          A lens made of glass of refractive index\n\n1.52 has focal length of 10 cm in air and 50\ncm when immersed in liquid. The\nrefractive index of liquid must be:\n[KU 2017]
          📅KU 2017
          Q68.
          The focal length of lens is F, and diameter\nof aperture is d. When y of diameter is\nblackened , then intensity of image will be:\n[IOM 2017]
          📅IOM 2017