29Refraction through Lenses

📚
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.
          A lens is immersed in liquid having refractive index same as lens. New focal length will be
          📅CEE MBBS 2025
          Q2.
          Converging lens f = 5 cm and diverging lens f = 10 cm are in contact. Power is
          📅CEE MBBS 2025
          Q3.
          The focal length of a convex lens depends upon:
          📅CEE MBBS 2024a
          Q4.
          Two thin lenses of focal lengths f₁ and f₂ are placed in contact coaxially. The combination is equivalent to a single lens of power:
          📅CEE MBBS 2024a
          Q5.
          If refractive index \(\mu=1.5\) and power is 4 D for an equiconvex lens, the radius of curvature is:
          📅CEE MBBS 2023
          Q6.
          The image formed by a convex lens when the object is at 2F is:
          📅CEE MBBS 2022
          Q7.
          Radius of curvature of a plano-convex lens is 10 cm and focal length is 30 cm. Refractive index of the lens material is:
          📅CEE MBBS 2022
          Q8.
          In the surrounding medium of a convex lens is replaced by water instead of air, the effect on the lens will be:
          📅CEE BDS 2024
          Q9.
          The focal length of a convex lens is f, when an object is placed between F and 2F, the image formed is:
          📅CEE BDS 2021
          Q10.
          The power of convex lens is P₁ = +6D and power of concave lens is P₂ = -4D. The focal length of combinations is:
          📅BP 2013
          Q11.
          A concave mirror has radius of curvature 40 cm and forms an image of magnification +2. Find the object distance.
          📅BP 2012
          Q12.
          Two thin lenses of focal length +60 cm and -20 cm are placed in contact. The focal length of combination is:
          📅BP 2011
          Q13.
          An object of height 1 m is placed at the principal focus of a convex lens. Where is its image formed?
          📅BP 2010
          Q14.
          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
          Q15.
          P and q are the distances of object and image from the principal focus of an equiconvex lens. Newton's formula for its focal length is:
          📅MOE 2012
          Q16.
          A biconvex lens of 8 cm and 12 cm radius of curvature with refractive index 1.5 has focal length:
          📅MOE 2011
          Q17.
          If a biconvex lens is silvered on one side, it will behave as:
          📅IOM 2013
          Q18.
          A convex lens has focal length 20 cm. Its power is:
          📅IOM 2012
          Q19.
          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
          Q20.
          Three thin lenses of focal lengths +10 cm, -5 cm and +30 cm are placed in contact. What is their equivalent focal length?
          📅IE 2013
          Q21.
          Two thin lenses (10 cm and 20 cm focal lengths) are placed in contact. Their combined focal length is:
          📅KU 2013, 2012
          Q22.
          Two lenses, one convex and one concave, have focal lengths +20 cm and -30 cm. What is the equivalent focal length when placed in contact?
          📅IE 2013
          Q23.
          A plano-convex lens has radius of curvature 10 cm and focal length 30 cm. Its refractive index is:
          📅MOE 2010
          Q24.
          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
          Q25.
          The distance between an object and a diverging lens is 'p' times the focal length. The lateral magnification 'm' is:
          📅IOM 06
          Q26.
          The focal length of a lens in air is 30 cm. In water (μ_{w} = 1.33, μ_{lens} = 1.5), its focal length is:
          📅IOM 02
          Q27.
          Two lenses of power +12D and -2D are placed in contact. The focal length of the combination is:
          📅MOE 06
          Q28.
          The effective power if lenses of focal length +10 cm and -20 cm are combined is:
          📅MOE 2056
          Q29.
          If an object is placed at the focus of a convex lens, the refracted rays are:
          📅MOE
          Q30.
          A glass convex lens has focal length 12 cm in air. It is immersed in water (assume refractive indices 1.5 for glass and 4/3 for water). Its focal length becomes approximately:
          📅Bangladesh Emb
          Q31.
          An object is placed left of a convex lens forming an image on a screen. If the screen is shifted away:
          📅KU 08
          Q32.
          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
          Q33.
          A convex lens is dipped in a liquid with refractive index equal to the lens. Its focal length becomes:
          📅IE-04
          Q34.
          A convex lens forms a real image at a distance 1.5 times its focal length from the lens. Where is the object?
          📅IE-05
          Q35.
          An object is 1 cm in front of a convex lens. It forms a virtual, erect image with magnification +5. Find the focal length.
          📅IE-06
          Q36.
          A converging lens forms an image 1.5f beyond the lens. The object is:
          📅TE-07
          Q37.
          An object is placed 10 cm in front of a diverging lens (f = -20 cm). The image will be:
          📅IE-01
          Q38.
          A plano-convex lens (μ = 1.5, R = 20 cm) has focal length:
          📅BPKIHS-08
          Q39.
          A plano-convex and plano-concave lens (radii R, refractive indices μ₁ and μ₂) have combined focal length:
          📅BPKIHS-09
          Q40.
          Two lenses (P = +1.75D and -1.25D) are combined. The focal length of the combination is:
          📅BPKIHS-97
          Q41.
          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
          Q42.
          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
          Q43.
          A convex lens produces a real image n times the size of the object. The object distance is:
          📅MOE
          Q44.
          A convex lens produces a virtual image n times the size of the object. The object distance is:
          📅MOE
          Q45.
          A concave lens has a focal-length magnitude f and gives an erect image one-nth the object height. What is the object distance magnitude?
          📅MOE
          Q46.
          A plano-convex lens (μ, radius R) is silvered on the plane side. The system behaves like a concave mirror of radius:
          📅MOE
          Q47.
          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
          Q48.
          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
          Q49.
          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
          Q50.
          An object is placed 20 cm from a convex lens (f = 10 cm). The image is formed at:
          📅MOE
          Q51.
          Two thin lenses (f₁, f₂) are placed at distance 'd'. For zero power, the separation 'd' is:
          📅MOE
          Q52.
          A convex lens of focal length f1 and a concave lens of focal-length magnitude f2 are in contact. When does the pair converge?
          📅MOE
          Q53.
          A convex lens (+6D) and concave lens (-4D) in contact form a combination with:
          📅MOE
          Q54.
          A plano-convex lens (R = 10 cm, f = 30 cm) has refractive index:
          📅MOE
          Q55.
          A double convex lens (μ = 1.5, R = 20 cm) converges parallel rays at a distance:
          📅MOE
          Q56.
          A lens (f in air, μ = 1.5) is placed in liquid (μ = 1.33). Its focal length becomes:
          📅MOE
          Q57.
          For a convex lens (f_{v}, f_{r}) and concave lens (F_{v}, F_{r}):
          📅MOE
          Q58.
          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
          Q59.
          A symmetric double convex lens (P = 4D) is cut into two equal parts. The power of each part is:
          📅MOE
          Q60.
          The focal length of a plano-convex lens in air equals the radius of curvature of its convex surface. Its refractive index is:
          📅MOE
          Q61.
          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
          Q62.
          A convex lens forms a 4 cm image on a screen. When shifted, it forms a 16 cm image. The object length is:
          📅MOE
          Q63.
          A convex lens forms images with magnifications 2 and 0.5 for two positions separated by 30 cm. Its focal length is:
          📅MOE
          Q64.
          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
          Q65.
          Which biconvex lens of refractive index 1.5 has the greatest power, if both surfaces have equal radii of curvature in magnitude?
          📅MOE
          Q66.
          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
          Q67.
          A concavo-convex lens (R₁ = 40 cm, R₂ = 20 cm, μ = 1.5) has focal length:
          📅MOE
          Q68.
          A camera with focal length 5 cm photographs a 5 km-wide stretch of terrain as a 5 cm image on the film. Estimate its altitude for a distant, nearly flat landscape.
          📅MOE
          Q69.
          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 ?
          Q70.
          A cyclist moves sideways at 10 m/s, 10 m from a camera lens of focal length 10 cm. What maximum exposure time limits image movement to 1 mm?
          Q71.
          A plano-convex lens has diameter 6 cm and central thickness 3 mm (plane side tangent to the rim). The light speed in glass is 2 × 10⁸ m/s. Find its approximate focal length.
          Q72.
          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
          Q73.
          If lens behaves as converging in air and\ndiverging in water. Then refractive index is\n[TOM 20151
          📅TOM 2015
          Q74.
          When the convex lens of refractive index\n(H), immersed in water of same refractive\nindex (1) then, its focal length is:
          📅KU 2016
          Q75.
          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
          Q76.
          A lens of aperture diameter d forms an image with intensity I. Its central circular portion of diameter d/2 is blackened. What is the new image intensity?
          📅IOM 2017