Assertion The formula connecting u, v and f for a spherical mirror is valid only for mirrors whose s

Q: Why do we use the paraxial approximation?

Without it, rays don't converge at a single point, making it impossible to define a single focal length or use a simple algebraic formula for image formation.

Q: What is spherical aberration?

It's an optical defect where marginal rays (far from the axis) focus closer to the mirror than paraxial rays, causing a blurred image.

Q: Do the laws of reflection fail for rough surfaces?

No! The laws of reflection hold true at every microscopic point of incidence. Rough surfaces cause diffuse reflection because the local normals point in different directions, not because the laws fail.

Q: Can we fix spherical aberration?

Yes, by using parabolic mirrors instead of spherical ones. A parabolic mirror focuses all parallel rays, paraxial or marginal, to a single precise point.

Q: Is the mirror formula completely useless for large mirrors?

It's not useless, but it becomes inaccurate for rays striking the outer edges. For the central region of a large mirror, it still works fine.

Question

Assertion The formula connecting $u$, $v$ and $f$ for a spherical mirror is valid only for mirrors whose sizes are very small compared to their radii of curvature.
Reason Laws of reflection are strictly valid for plane surfaces, but not for large spherical surfaces.

Options:

(A) If Assertion is true, Reason is true; Reason is the correct explanation for Assertion.

(B) If Assertion is true, Reason is true; Reason is not a correct explanation for Assertion.

(C) If Assertion is true; Reason is false.

(D) If Assertion is false; Reason is true.

Sigma Percentile · Accepted Answer

### The Paraxial Approximation

Have you ever wondered why the mirror formula, $$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$, is so beautifully simple? It almost feels too perfect. Well, that's because it comes with a hidden catch: the **paraxial approximation**.

When we derive this formula, we assume that all light rays strike the mirror very close to the principal axis. These rays are called **paraxial rays**. Because they hit the mirror at very small angles, they all converge neatly at a single, sharp point—the focus ($F$).

### The Curse of Spherical Aberration

But what happens if we use a massive spherical mirror? Rays that strike the mirror far from the principal axis are called **marginal rays**. Because of the spherical geometry, these marginal rays bend too sharply and cross the principal axis *closer* to the mirror than the paraxial rays do.

This means the light doesn't focus at a single point anymore! The result is a blurred, distorted image. This optical defect is known as **spherical aberration**. To avoid this and keep our simple formula accurate, we must ensure the mirror's aperture (its size) is very small compared to its radius of curvature. Thus, the **Assertion is absolutely true**.

### The Universality of Reflection

Now, let's look at the Reason. It claims that the laws of reflection ($\angle i = \angle r$) only work for plane surfaces and fail for large spherical ones.

This is a fundamental misunderstanding of physics! The laws of reflection are a local phenomenon. Imagine zooming in infinitely close to the point where a light ray hits a curved mirror. At that microscopic level, the surface looks perfectly flat. If you draw a tangent and a normal at that exact point, the angle of incidence will *always* equal the angle of reflection.

The laws of reflection never fail, whether the surface is a flat mirror, a giant spherical telescope, or even a crumpled piece of aluminum foil. Therefore, the **Reason is completely false**.

Select Answer:

Visualized Solution

\text{Analyzing the Assertion}

\text{Paraxial Rays}

\text{Marginal Rays \& Spherical Aberration}

\text{Analyzing the Reason}

\text{Universality of Laws of Reflection}

\text{Conclusion}

The Sigma Insight: Spherical Mirror

The Paraxial Approximation

The Curse of Spherical Aberration

The Universality of Reflection

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