Problem 70 - Master Resource Book in JEE Main Physics

Question

A concave mirror and a converging lens (glass with \(\mu=1.5\) ) both have a focal length of \(3 \mathrm{~cm}\) when in air. When they are in water \(\left(\mu=\frac{4}{3}\right)\), their new focal length are (a) \(f_{\text {Lem }}=12 \mathrm{~cm}, f_{\text {sins }}=3 \mathrm{~cm}\) (b) \(f_{\text {Lens }}=3 \mathrm{~cm}, \overline{f_{\text {Miner }}}=12 \mathrm{~cm}\) (c) \(f_{\text {Lens }}=3 \mathrm{~cm}, \overline{f_{\text {Miner }}}=3 \mathrm{~cm}\) (d) \(f_{\text {Lmi }}=12 \mathrm{~cm}, f_{\text {uinar }}=12 \mathrm{~cm}\)

Step-by-Step Solution

Answer

Option (c): Both the lens and the mirror have a focal length of 3 cm in water.

Step 1: Understand the Question

We need to find the new focal lengths of a concave mirror and a converging lens when placed in water. The refractive index (\(\mu\)) changes as the mediums change from air to water.

Step 2: Apply Lens Formula in Air

The focal length of a lens in air (\(f_{\text{air}}\)) is given, which is 3 cm. The refractive index of glass is \(\mu_{g} = 1.5\).

Step 3: Calculate New Focal Length of the Lens in Water

The lens-maternal formula is \(\frac{f_{\text{water}}}{f_{\text{air}}} = \frac{\mu_{\text{air}}-1}{\mu_{\text{lens}}-\mu_{\text{water}}}\). Substituting \(\mu_{\text{air}} = 1\), \(\mu_{\text{water}} = \frac{4}{3}\), and \(\mu_{\text{lens}} = 1.5\), we solve: \[ f_{\text{water}} = f_{\text{air}} \times \frac{1-1}{1.5- \frac{4}{3}} = 3 \times \frac{1- \frac{4}{3}}{1.5- \frac{4}{3}} \]This results in the focal length remaining unchanged at 3 cm.

Step 4: Analyze the Concave Mirror's Focal Length

The focal length of a mirror does not depend on the medium, so it remains the same regardless. The focal length of the concave mirror in water will stay at 3 cm.

Step 5: Conclusion and Match with Options

From the calculations, both the lens and the mirror have the same focal length of 3 cm in water. The correct option with \(f_{\text{Lens}} = 3\text{ cm}\) and \(f_{\text{Mirror}} = 3\text{ cm}\) is the option (c).

Key Concepts

Concave Mirror

A concave mirror, often known as a converging mirror, has a reflective surface that curves inward. This inward curve creates a focal point where parallel rays of light converge after reflection.
When rays parallel to the principal axis strike a concave mirror, they are reflected to meet at a point called the focus (F). The distance from the mirror surface to the focus is known as the focal length (f).
- Concave mirrors can form real images when the object is placed beyond the focal point. - If the object is placed at the focal point, the reflected rays will be parallel and the image is formed at infinity.
The characteristic of the mirror's focal length is independent of the surrounding medium, as seen in the original exercise. Hence, whether in water or air, the focal length of the concave mirror remains consistent.

Converging Lens

A converging lens, also referred to as a convex lens, is thicker in the middle and thinner at the edges. This shape causes parallel incoming light rays to bend inward towards the focal point on the other side of the lens.
The ability of a lens to converge light and form images is precisely what defines its focal length.
- For converging lenses, real and inverted images are produced when the object is placed beyond the focal length. - A virtual, upright, and magnified image is formed when the object is placed between the focal point and the lens.
In the initial scenario, the behavior of converging lenses is altered by the medium they are in, unlike mirrors. Consequently, the formula for adjusting the focal length when lenses are moved from air to water becomes essential.

Refractive Index

The refractive index (\(\mu\)) of a material defines how much it bends (or refracts) light. It compares the speed of light in the material to its speed in a vacuum, which is always the standard (or 1).
Refractive index plays a key role when light travels from one medium to another, especially in lenses.

• For glass, the refractive index is higher, implying that light travels more slowly through it.
• Water has a refractive index of \(\mu=\frac{4}{3}\) and slows down light but less than glass does.

Using the refractive index helps in calculating how lenses behave in different environments, as shown by the lens formula applied in the original solution.

Focal Length

The focal length of a lens or mirror is a critical measure in optics that refers to the distance over which parallel rays of light either converge to a single point or appear to diverge.
For lenses:

• The focal length is positive for converging lenses.
• Shorter focal lengths indicate stronger converging power.

A mirror's focal length, especially for spherical mirrors, remains constant regardless of the medium it is in, while a lens's focal length may change depending on the surrounding refractive environment.
In the exercise, it was calculated that the concave mirror's focal length didn't change, while recalculating the lens's focal length involved considering the surrounding medium.

Lens Formula

The lens formula is an expression that relates the focal length of a lens to the object and image distances. It is written as \(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\), where:

• \(f\) represents the focal length.
• \(v\) is the image distance from the lens.
• \(u\) is the object distance from the lens.

In a more complex calculation involving different mediums, the lens-maternal formula becomes important.
The adjusted lens formula in the presence of a new medium, as seen in the example, has to factor in differing refractive indices to ensure accurate results.
This formula allows you to understand and predict how a lens will focus light when transitioning between substances such as air and water.