LIGHT:REFLECTION AND REFRACTION-IV : NUMERICALS

SOLVED NUMERICALS OF TEXT BOOK (GSEB)

*INDEX-TOPIC SEARCH

1. If an object of height 4 cm is placed at distance of 12 cm from a concave mirror having focal length 24 cm, find the position, nature and the height of the image.
   
   SOLUTION
   
   Here,
   
   object height h = 4 cm
   
   object dis tance u = -12 cm
   
   focal length f = -24 cm
   
   Putting these values in mirror formula,
   
   (1/u) + (1/v) = (1/f)
   
   ∴ (-1/12) + (1/v) = (-1/24)
   
   ∴ (1/v) = (1/12) - (1/24)
   
   ∴ (1/v) = (1/24)
   
   ∴ v = 24
   
   Now, magnification m = (-u/v) = (-24)/(-12) = 2
   
     ∴ (h'/h) = 2
   
     ∴ h' = 2h = 2 x 4 = 8 cm
   
   Since v is positive, the image is formed behind the mirror at the distance of 24 cm from the mirror.It is virtual and has height of 8 cm.
   


2. An object of height 6 cm is placed at a distance of 10 cm from a convex mirror with radius of curvature 30 cm.Find the position, nature and the height of its image.
   
   SOLUTION
   
   Here,
   

   Object distance u = -10 cm
   Radius of curvature R = 30 cm
   Object height h = 6 cm

   
   
   Now,
   
   focal length f = (R/2) = 30/2 = +15
   
   Using mirror formula,
   
    (1/u) + (1/v) = (1/f)
   
     ∴ (1/v) = (1/f) - (1/u)
   
     ∴ (1/v) = (1/15) - (1/-10)
   
     ∴ (1/v) = (1/15) + (1/10) =(1/6)
   
     ∴ v = 6 cm
   
   Also, magnification m = -(u/v) = -(6/-10) = 0.6
   
     ∴ (h'/h) = 0.6
   
     ∴ h' = mh =0.6 x 6 = 3.6 cm
   
   Thus, the image is formed behind the mirror(as v is positive), it is virtual and its height is 3.6 cm.
   


3. The refractive index of water with respect to air is 1.33. The velocity of light in vacuum is 3 x 10⁸ m/s. Calculate the velocity of light in water.
   
   SOLUTION
   
   Here, the refractive index of water with respect to air is 1.33.
   
     ∴ absolute refractive index of water = 1.33
   
   Now, Absolute refractive index of water =
   

   Velocity of light in vacuum
   Velocity of light in water
   

   
   
     ∴ Velocity of light in water =
   

   Velocity of light in vacuum
   Absolute refractive index of water
   
   = 3 x 10⁸
   1.33
   
   = 2.25 x 10⁸ m/s.
   

   
   


4. A ray of light enters from water to glass.Refractive index of glass with respect to water is 1.12. Find absolute refractive index of water if absolute refractive index of glass is 1.5.
   
   SOLUTION
   
   Taking water as "medium1" and glass as "medium2",
   
   Absolute refractive index of water, η₁ = ?
   
   Absolute refractive index of glass, η₂ = 1.5
   
     η₂₁ = 1.12_________(given)
   
   ∴ η₂/η₁ = 1.12
   
   ∴ η₁ =η₂/1.12
   
   ∴ η₁ = (1.5)/(1.12) = 1.34
   
   Hence,absolute refractive index of water = 1.34
   


5. A swimmer lights a torch under sea water.Light from the torch is incident on water surface in such a way that incident light makes an angle of 37° with water surface.Find the angle of refraction if absolute refractive indices of water and air are 1.33 and 1.0 respectively.
   
   SOLUTION
   
   Taking water as "medium1" and air as "medium2",we have
   
     η₁ = 1.33
   
     η₂ = 1.00
   
   Angle of incidence, θ₁ = 37°
   
   Angle of refraction, θ₂ = ?
   
   Now, (sin θ₁/sin θ₂) = (η₂/η₁)
   
   ∴ (sin 37°/sin θ₂) = (1.00/1.33)
   
   ∴ sin θ₂ = [ (0.6015 x 1.33)/1.00]
   
   ∴ sin θ₂ = 0.8000
   
   ∴ θ₂ = 53°_____(from the table of sines)
   
   Hence, angle of refraction = 53°
   


6. An object is placed at 30 cm in front of a convex lens of focal length 20 cm. Find the position of the image.
   
   SOLUTION
   
   Here,
   
   Focal length f = +20 cm
   
   Object distance u = -30 cm
   
   Image distance v = ?
   
   Using lens formula,
   
     (1/f) = (1/v) - (1/u)
   
   ∴ (1/v) = (1/u) + (1/f)
   
   ∴ (1/v) = (1/-30) + (1/20)
   
   ∴ (1/v) = (-2 + 3)/60
   
   ∴ (1/v) = (1/60)
   
   ∴ v = 60 cm.
   
   The positive value of v indicates that the image is formed at 60 cm on the right side of lens from the center of the lens.
   


7. At what distance the object should be placed so that the image will be formed at a distance 10 cm from a concave lens ? Focal length of the lens is 20 cm.
   
   SOLUTION
   
   Since the lens is concave, the image will be formed on the same side as that of the object.
   
   Here,
   
   Image distance v = -10 cm
   
   Focal length f = -20 cm
   
   Object distance u = ?
   
   Using lens formula,
   
     (1/f) = (1/v) - (1/u)
   
   ∴ (1/u) = (1/v) - (1/f)
   
   ∴ (1/u) = (1/-10) - (1/-20)
   
   ∴ (1/u) = (-2 + 1)/20
   
   ∴ (1/u) = -(1/20)
   
   ∴ u = -20 cm
   
   Thus the object should be placed 20 cm from the concave lens on the left side of the lens.
   

UNSOLVED NUMERICALS OF TEXT BOOK (GSEB)

1. An object of 5 cm height is placed at a distance of 15 cm from a concave mirror.Find the position, height and nature of its image.The focal length of the mirror is 10 cm.
   
   SOLUTION
   
   Here
   
   height of object, h = 5 cm
   
   Distance of object, u = -15 cm
   
   Focal length of concave mirror, f = -10 cm
   
   Distance of image, v = ?
   
   Using mirror formula,
   
     (1/v) + (1/u) = (1/f)
   
   ∴ (1/v) = (1/f) - (1/u)
   
   ∴ (1/v) = (1/-10) - (1/-15)
   
   ∴ (1/v) = -(1/10) + (1/15)
   
   ∴ (1/v) = (-3 + 2)/30 = -(1/30)
   
   ∴ v = -30 cm
   
   Also,(image height)/(object height) = (-v/u)
   
   ∴ (h'/h) = (-v/u)
   
   ∴ (h'/5) = (-30/-15)
   
   ∴ h' = 10 cm
   
   Hence the image is formed at a distance of 30 cm ; its height is 10 cm and it is real and inverted.
   


2. An object of 10 cm height is placed at a distance of 10 cm from a convex mirror.The radius of curvature of the mirror is 30 cm.Find the position, height and nature of its image.
   
   SOLUTION
   
   Here
   
   Height of object, h = 10 cm
   
   Distance of object, u = -10 cm
   
   Radius of curvature of convex mirror, R = 30 cm
   
   Distance of image, v = ?
   
   Since f = (R/2),
   
     f = (30/2) = 15 cm
   
   Using mirror formula,
   
     (1/u) + (1/v) = (1/f)
   
   ∴ (1/v) = (1/f) - (1/u)
   
   ∴ (1/v) = (1/15) - (1/-10) = (1/15) + (1/10)
   
   ∴ (1/v) = (2 + 3)/30 = (1/6)
   
   ∴ v = 6 cm
   
   Also,(image height)/(object height) = (-v/u)
   
   ∴ (h'/h) = (-v/u)
   
   ∴ (h'/10) = (-6/-10)
   
   ∴ h' = 6 cm
   
   Hence the image is formed at a distance of 6 cm; its height is 6 cm and it is virtual and erect.
   


3. Rays of light are entering from glass to glycerin.If the absolute refractive index of glass is 1.5 and that of glycerin is 1.47 then find the refractive index of glycerin with respect to glass.
   
   SOLUTION
   
   Let's consider glass as medium1 and glycerin as medium2.
   
   Absolute refractive index of glass, η₁ = 1.5
   
   Absolute refractive index of glycerin, η₂ = 1.47
   
   Now,
   
   Refractive index of glycerin with respect to glass = η₂₁
   
                 =η₂/η₁ = 1.47/1.5 = 0.98
   
   Hence the refractive index of glycerin with respect to glass is 0.98.
   


4. Rays of light are entering from air to water. If the angle of incidence at the surface separating two mediums is 70°, find the angle of refraction of light in water.Absolute refractive index of water is 1.33.
   
   SOLUTION
   
   Let's consider air as medium1 and water as medium2.
   
   Absolute refractive index of air, η₁ = 1.00
   
   Absolute refractive index of water, η₂ = 1.33
   
   Angle of incidence θ₁ = 70°
   
   Angle of refraction θ₂ = ?
   
   Now, sin θ₁/sin θ₂ = η₂/η₁
   
   ∴ (sin 70°/sin θ₂) = (1.33/1.00)
   
   ∴ (0.9397/sin θ₂) = (1.33/1.00)
   
   ∴ sin θ₂ = 0.7065
   
   ∴ θ₂ = 44° 57'
   
   Hence the angle of refraction = 44° 57'
   


5. An object is placed perpendicular to the principal axis of a convex lens of focal length 10 cm at a distance of 20 cm from the lens. Find the position of the image.
   
   SOLUTION
   
   Here
   
   Focal length of convex lens, f = 10 cm
   
   Distance of object, u = -20 cm
   
   Distance of image, v = ?
   
   Using lens formula,
   
      (1/f) = (1/v) - (1/u)
   
   ∴ (1/v) = (1/f) + (1/u)
   
   ∴ (1/v) = (1/10) + (1/-20) = (1/10) - (1/20)
   
   ∴ (1/v) = (2 - 1)/20 = (1/20)
   
   ∴   v = 20 cm
   
   Hence the image is formed at the distance of 20 cm from the lens.
   


6. An object is placed perpendicular to the principal axis of a concave lens of focal length 30 cm at a distance of 20 cm from the lens. Find the position of the image.
   
   SOLUTION
   
   Here
   
   Focal length of concave lens, f = -30 cm
   
   Distance of the object, u = -20 cm
   
   Distance of the image, v = ?
   
   Using lens formula,
   
   
      (1/f) = (1/v) - (1/u)
   
   ∴ (1/v) = (1/f) + (1/u)
   
   ∴ (1/v) = (1/-30) + (1/-20) = (-2 - 3)/60
   
   ∴ (1/v) =(-5/60) = (-1/12)
   
   ∴ v = (-12) cm
   
   Hence the image is formed at the distance of 12 cm from the lens on the same side of the object.
   


7. Find the power of a convex lens with focal length 0.4 m
   
   SOLUTION
   
   Power of lens, p = (1/f)
   
   ∴ p = (1/0.4) ______________(for convex lens f is +ve)
   
   ∴ p = 2.5 D
   
   Hence the power of lens is 2.5 D (dioptre).