A convex mirror has a focal length f when it is in air. If this mirror is completely immersed in a liquid of refractive index 4/3, what will be its focal length in the liquid?

Introduction / Context:

This question is about spherical mirrors and the effect of surrounding medium on focal length. Students often learn that the focal length of a lens depends on refractive indices of the lens material and the surrounding medium. However, mirrors form images by reflection, not refraction, and therefore behave differently. Understanding this distinction helps avoid common conceptual mistakes.

Given Data / Assumptions:

• A convex (diverging) spherical mirror has focal length f in air.
• The mirror is then immersed in a liquid of refractive index 4/3 relative to air.
• The mirror material and its radius of curvature remain unchanged.
• We are asked for the new focal length of the mirror in the liquid.

Concept / Approach:

The focal length of a spherical mirror depends only on its radius of curvature and the geometry of reflection. For a mirror, the approximate relation is f = R / 2, where R is the radius of curvature. This relation arises purely from reflection laws (angle of incidence equals angle of reflection) and the mirror shape. Unlike lenses, mirrors do not depend on the refractive index of the surrounding medium because light does not refract through the mirror; it reflects from the mirror surface. Therefore, immersing a mirror in a different medium does not change its focal length.

Step-by-Step Solution:

Verification / Alternative check:

Contrast this with lenses: the focal length of a lens depends on the refractive index ratio between the lens material and its surroundings, as described by the lens maker formula. If a lens is immersed in a liquid whose refractive index is close to that of the lens, the focal length changes significantly. For mirrors, any ray diagram you draw in different media still uses the same reflection angles and curvature, leading to identical convergence or divergence behaviour. This thought experiment confirms that the focal length remains unchanged for a mirror.

Why Other Options Are Wrong:

• (4/3)f: Suggests scaling of focal length with refractive index, which would apply to lenses, not mirrors.
• (3/4)f: Similarly implies dependence on the index in the opposite direction, which is incorrect for mirrors.
• (7/3)f: Has no standard basis in mirror optics and is clearly unrelated to the simple relation f = R / 2.

Common Pitfalls:

The main pitfall is confusing the behaviour of mirrors with that of lenses. Many students automatically assume that any optical element will have its focal length changed by immersion in a liquid. This is true for refracting elements like lenses but not for reflecting elements like mirrors. Another error is to think that the speed of light in the medium alone determines focal length; for mirrors, the path geometry and reflection law dominate. Always remember: mirrors depend on shape and reflection, lenses depend on shape and refractive indices.

Final Answer:

The focal length of the convex mirror in the liquid remains f.