Problem 55 - Master Resource Book in JEE Main Physics

Question

A plano-convex lens of refractive index \(1.5\) and radius of curvature \(30 \mathrm{~cm}\) is silvered at the curved surface. Now this lens has been used to form the image of an object. At what distance from this lens an objeet be placed in order to have a real image of the size of the object? (a) \(20 \mathrm{~cm}\) (b) \(30 \mathrm{~cm}\) (c) \(60 \mathrm{~cm}\) (d) \(80 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The object should be placed at \(60\text{ cm}\) from the lens, choice (c).

Step 1: Understand the Configuration

The problem involves a plano-convex lens silvered on the curved side, forming a lens-mirror system. The lens has a refractive index \( n = 1.5 \) and radius of curvature \( R = 30 \text{ cm} \). The lens will act as a plano-concave mirror due to silvering.

Step 2: Calculate Focal Length of the Lens

To find the effective focal length of this combined system, first find the focal length of the lens using the lens maker's formula: \[ \frac{1}{f_{lens}} = \left(n - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]Here, \( R_1 = \infty \) for the plane surface and \( R_2 = -30 \text{ cm} \) for the curved surface. Hence,\[ \frac{1}{f_{lens}} = (1.5 - 1) \left( \frac{1}{\infty} + \frac{1}{30} \right) = \frac{0.5}{30} = \frac{1}{60} \]Thus, \( f_{lens} = 60 \text{ cm} \).

Step 3: Calculate Focal Length of the Silvered System

After silvering, the lens acts as a concave mirror with focal length \( f_{mirror} = -\frac{R}{2} \).Using \( R = 30 \text{ cm} \), \[ f_{mirror} = -15 \text{ cm} \].To find the focal length of the lens-mirror system, use:\[ \frac{1}{f_{combined}} = \frac{2}{f_{lens}} + \frac{1}{f_{mirror}} \]\[ \frac{1}{f_{combined}} = \frac{2}{60} - \frac{1}{15} = \frac{1}{30} - \frac{1}{15} = -\frac{1}{30} \]Thus, \( f_{combined} = -30 \text{ cm} \).

Step 4: Determine Object Distance for Real Image

For a real image of the same size as the object, the object must be placed at the center of curvature (radius of curvature) of the effective mirror. Since this combined system has a focal length of \(-30 \text{ cm}\), the object should be placed at the radius of curvature \(2f\), which is \(-60 \text{ cm}\) from the silvered surface.

Key Concepts

Lens-Mirror System

Understanding a lens-mirror system can be tricky, yet it's a fascinating application in optics. This system combines both a lens and a mirror in one setup, often to enhance image formation or to achieve specific optical effects. In this case, the plano-convex lens acts as a lens-mirror system due to silvering on its curved side.
Silvering transforms the curved side of the lens into a mirror. This alters the way light interacts with the system. As light passes through the glass and reflects off the silver, it behaves as though it's passing through a combination of a lens and a mirror.

• The lens forms images by refracting light, the bending, and focusing of rays.
• The mirror forms images by reflecting light back through the lens.

This hybrid system can often be reduced to a single focal length for practical imaging purposes. It simplifies understanding but ensures that both lens and mirror effects are considered.

Plano-Convex Lens

A plano-convex lens is a lens with one flat (plane) surface and one convex (curved outward) surface. This configuration is pivotal in focusing light. Due to its design, it converges parallel light rays toward a focal point, making it widespread in focusing applications.
The lens's behavior is particularly influenced by its:

• Refractive index ( 1.5 for the problem at hand), indicating how much it bends light.
• Radius of curvature (30 cm in this case), determining the shape of the curved surface.

The plano side is of infinite radius, having no curvature, which simplifies calculations. In optics, this lens type is prized for forming clear, well-defined images, essential in optical instruments like cameras and microscopes.

Focal Length Calculation

The focal length () of an optical system is essentially the distance at which parallel light rays converge or appear to diverge. Calculating it precisely is crucial for understanding the imaging properties. For a simple lens, it is determined by the lens maker's formula:\[\frac{1}{f_{lens}} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)\]Here, the important parameters include:

• Refractive index (), influencing how strongly the lens refracts light.
• Radii of curvature ( and ).

For our scenario, after silvering, the system functions differently. By applying the focal length formula, we find both the lens and mirror's contributions to derive the system's focal length:

• First, calculate the focal length of the lens before silvering.
• Then, adjust for the mirror effects post-silvering.
• Combine effects to find the effective focal length of the complete lens-mirror system.

Understanding these calculations gives insight into placing objects in the optimal position for desired image sizes and properties.