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A plano-convex lens is made of a material of refractive index n. When a small object is placed 30 cm away
in front of the curved surface of the lens, an image of double the size of the object is produced. Due to
reflection from the convex surface of the lens, another faint image is observed at a distance of 10 cm away
from the lens. Which of the following statement(s) is(are) true?
  • a)
    The refractive index of the lens is 2.5
  • b)
    The faint image is erect and real
  • c)
    The focal length of the lens is 20 cm
  • d)
    none of these
Correct answer is option 'A,C'. Can you explain this answer?
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A plano-convex lens is made of a material of refractive index n. When ...
For refraction through lens,
The faint image is erect and virtual.
 
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A plano-convex lens is made of a material of refractive index n. When ...
Given information:
- A plano-convex lens is made of a material of refractive index n.
- A small object is placed 30 cm away in front of the curved surface of the lens.
- An image of double the size of the object is produced.
- Due to reflection from the convex surface of the lens, another faint image is observed at a distance of 10 cm away from the lens.

To determine:
Which of the following statement(s) is(are) true?

a) The refractive index of the lens is 2.5
b) The faint image is erect and real
c) The focal length of the lens is 20 cm
d) none of these

Explanation:
Statement a): The refractive index of the lens is 2.5
- Let's assume the object distance (u) is positive and the image distance (v) is positive when the image is formed on the opposite side of the object.
- According to the lens formula, 1/f = 1/v - 1/u, where f is the focal length of the lens.
- In this case, the image formed is double the size of the object, which means the magnification (m) is given by m = -v/u = 2.
- Using the magnification formula, m = -v/u = h'/h, where h' is the height of the image and h is the height of the object.
- Since the magnification is positive, both h' and h should have the same sign.
- Therefore, the image is erect.
- The formula for the magnification in terms of the object and image distances is given by m = -v/u.
- Substituting the given values, m = -v/30 = 2, we can solve for v to find v = -60 cm.
- The image distance is negative, indicating that the image is formed on the same side as the object.
- The lens formula then becomes 1/f = 1/-60 - 1/30 = -1/60 - 2/60 = -3/60 = -1/20.
- Solving for f, we find f = -20 cm.
- The negative sign indicates that the lens is a converging lens, which means the refractive index should be greater than 1.
- Therefore, statement a) is true.

Statement c): The focal length of the lens is 20 cm
- As explained above, the focal length of the lens is -20 cm.
- Therefore, statement c) is true.

Conclusion:
- From the analysis, we can conclude that statement a) and c) are true.
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Most materials have the refractive index, n > 1. So, when a light ray from air enters a naturally occurring material, then by Snell’slaw,it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive index of the medium is given by the relation,n= c /where c is the speed of electromagnetic waves in vacuum, v its speed in the medium, εr and μr are the relative permittivity and permeability of the medium respectively.In normal materials, both εr and μr , are positive, implying positive n for the medium. When both εr and μr are negative, one must choose the negative root of n. Such negative refractive index materials can now be artificially prepared and are called metamaterials. They exhibit significantly different optical behavior, without violating any physical laws. Since n is negative, it results in a change in the direction of propagation of the refracted light.However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials. Q.Choose the correct statement.

Paragraph for Questions 13 and 14Most materials have the refractive index, n > 1. So, when a light ray from air enters a naturally occurringmaterial, then by Snell’s law, it is understood that the refracted ray bends towards the normal.But it never emerges on the same side of the normal as the incident ray. According to electromagnetism,the refractive index of the medium is given by the relation, where c is the speed ofelectromagnetic waves in vacuum, v its speed in the medium, εr and μr are the relative permittivity andpermeability of the medium respectively.In normal materials, both εr and μr, are positive, implying positive n for the medium. When both εr and μrare negative, one must choose the negative root of n. Such negative refractive index materials can nowbe artificially prepared and are called meta-materials. They exhibit significantly different optical behavior,without violating any physical laws. Since n is negative, it results in a change in the direction ofpropagation of the refracted light. However, similar to normal materials, the frequency of light remainsunchanged upon refraction even in meta-materials.Q.Choose the correct statement.

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A plano-convex lens is made of a material of refractive index n. When a small object is placed 30 cm awayin front of the curved surface of the lens, an image of double the size of the object is produced. Due toreflection from the convex surface of the lens, another faint image is observed at a distance of 10 cm awayfrom the lens. Which of the following statement(s) is(are) true?a)The refractive index of the lens is 2.5b)The faint image is erect and realc)The focal length of the lens is 20 cmd)none of theseCorrect answer is option 'A,C'. Can you explain this answer?
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