Understanding the Focal Length of a Convex Mirror in Liquid

The question asks about the focal length of a convex mirror when it is immersed in a liquid compared to its focal length in air. We are given the focal length in air as \(f\) and the refractive index of the liquid as \(\mu = \frac{4}{3}\).

What Determines the Focal Length of a Mirror?

The focal length of a spherical mirror, whether convex or concave, depends solely on its radius of curvature. The relationship between the focal length (\(f\)) and the radius of curvature (\(R\)) for a spherical mirror is given by:

\[f = \frac{R}{2}\]

For a convex mirror, the radius of curvature \(R\) is determined by the physical shape of the mirror's reflecting surface. It is the radius of the sphere from which the mirror is a part.

How Immersion in a Medium Affects Optics

When an optical instrument is placed in a different medium, its behaviour can change. This change is significant for lenses, whose action relies on the refraction (bending) of light as it passes from one medium (like air) into the lens material and then out into another medium (like liquid).

The focal length of a lens is given by the lens maker's formula, which explicitly includes the refractive indices of the lens material and the surrounding medium:

\[\frac{1}{f_{lens}} = (\mu_{lens, relative} - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)\]

Where \(\mu_{lens, relative}\) is the refractive index of the lens material relative to the surrounding medium. If the surrounding medium changes, \(\mu_{lens, relative}\) changes, and thus the focal length of the lens changes.

Why a Mirror's Focal Length Remains Unchanged in Liquid

Unlike lenses, mirrors work based on the principle of reflection, not refraction. Light rays strike the reflecting surface of the mirror and bounce back into the same medium. The process of reflection depends on the angle of incidence and the angle of reflection, which are governed by the law of reflection (\(\theta_i = \theta_r\)). This law is independent of the medium in which the reflection occurs.

The focal length of a mirror, as stated earlier, depends only on its radius of curvature \(R\). When a mirror is immersed in a liquid, its physical shape does not change. Therefore, its radius of curvature \(R\) remains the same.

Since the focal length \(f = R/2\) and \(R\) does not change when the mirror is immersed in the liquid, the focal length \(f\) of the mirror also remains unchanged.

Conclusion

The focal length of a convex mirror depends only on its geometry (specifically, its radius of curvature). Immersion in a liquid medium does not alter the mirror's shape or radius of curvature. Therefore, the focal length of the convex mirror remains the same in the liquid as it was in air.

Given that the focal length in air is \(f\), the focal length of the mirror in the liquid with refractive index \(\mu = \frac{4}{3}\) will also be \(f\).

Revision Table: Mirror vs. Lens in Different Media

Here is a quick comparison:

Additional Information: Types of Mirrors and Focal Length

Spherical mirrors are either concave or convex. Both types have a focal length related to their radius of curvature. The focal point is a point where parallel rays converge (for a concave mirror) or appear to diverge from (for a convex mirror) after reflection.

• Concave Mirror: Converging mirror. Focal length is usually considered positive in ray diagrams following certain sign conventions, or it is the distance from the vertex to the focal point. Real focal point.
• Convex Mirror: Diverging mirror. Focal length is usually considered negative in ray diagrams, or it is the distance from the vertex to the virtual focal point behind the mirror. Virtual focal point.

The fact that the focal length of a mirror is independent of the surrounding medium is a key difference when studying optics and comparing mirrors and lenses. The refractive index of the liquid (\(\mu = \frac{4}{3}\) in this question) is relevant for lenses but not for mirrors.