Problem 64 - Master Resource Book in JEE Main Physics

Question

A beam of parallel rays is brought to a focus by a plano-convex lens. A thin concave lens of the same focal length is joined to the first lens. The effect of this is (a) the focal points shifts away from the lens by a small distance (b) the focus remains undisturbed (c) the focus shifts to infinity (d) the focal points shifts towards the lens by a small distance

Step-by-Step Solution

Answer

The focus shifts to infinity (option c).

Step 1: Understanding Lens Properties

A plano-convex lens converges parallel rays of light to a focal point on the convex side. The focal length (\(f_1\) is positive for a converging lens. A thin concave lens, on the other hand, diverges rays and has a negative focal length (\(f_2\)).

Step 2: Calculating the Effective Focal Length

The focal length of two lenses in contact can be found using the formula: \[ \frac{1}{f_{eff}} = \frac{1}{f_1} + \frac{1}{f_2} \]Since both lenses have equal but opposite focal lengths, \(f_1 = -f_2\), and thus \(\frac{1}{f_1} + \frac{1}{f_2} = 0\).

Step 3: Determining the Position of the Focus

Since \(\frac{1}{f_{eff}} = 0\), the effective focal length (\(f_{eff}\)) approaches infinity. When the effective focal length goes to infinity, parallel rays do not converge to a point after passing through the lens. Instead, they exit the lens parallel to each other, which means the focus is now at infinity.

Key Concepts

Plano-convex Lens

A plano-convex lens is a simple optical device with one flat side and one convex side. It is designed to converge light rays that are initially parallel. When these rays pass through the lens, they bend towards the thicker part and meet at a single point known as the focal point.
One of the fascinating features of a plano-convex lens is its positive focal length. This value describes the distance from the lens to the focal point on the convex side. If you imagine holding a magnifying glass under the sun, the tiny focused dot of light on the paper below is an example of how a plano-convex lens works.

• The lens has a higher optical power, meaning it can bend light strongly.
• The curvature affects how much the light is bent - more curvature results in a shorter focal length.
• It is often used in applications like lighthouses and telescopes to focus light.

To better understand how a plano-convex lens functions, think about how it is thicker at the center than at the edges; this shape allows it to converge light effectively.

Concave Lens

A concave lens is quite the opposite of a convex lens in terms of how it handles light. It is thinner at the center and thicker at the edges. This shape causes light rays to diverge, or spread outwards. Thus, when parallel rays enter a concave lens, they appear to originate from a focal point located behind the lens.
With its negative focal length, a concave lens does not bring light rays to a real focus but instead creates what we call a virtual focus.

• A concave lens is commonly used in glasses for people who are nearsighted.
• It has lower optical power compared to convex lenses, meaning it bends light less.
• Despite diverging light, it is essential in correcting certain types of vision problems.

In practical terms, this lens is integral to optical instruments needing image reduction, like peepholes in doors.

Focal Length

The focal length of a lens is a crucial aspect of how lenses behave and determine their functionality. It is the distance from the lens to the focal point, the spot where light rays converge (or appear to diverge from in the case of a concave lens). Consider it as an indicator of how strongly a lens can bend light.
The formula for the effective focal length when combining two lenses of different types is given by:\[ \frac{1}{f_{eff}} = \frac{1}{f_1} + \frac{1}{f_2} \]Here, both lenses should be taken into account—each brings its own focal length in the calculation process.

• A positive focal length signifies a converging lens, like the plano-convex lens.
• A negative focal length, however, refers to a diverging lens, like the concave lens.
• When combined equally in magnitude but opposite in sign, they can nullify each other's effect, leading to an infinite effective focal length.

This concept of combining lenses is vital in various optical setups, showcasing how different lenses can manipulate light paths to achieve desired outcomes.