Given below are two statements:
One is labelled as Assertion A and the other is labelled as Reason R

Q: What exactly is resolving power?

Resolving power is the ability of an optical instrument to distinguish two closely spaced objects as separate entities. If they are closer than the resolution limit, they blur into a single blob.

Q: Why does wavelength limit resolution?

Because of diffraction. When waves pass through an aperture (like a lens), they spread out. The larger the wavelength compared to the aperture, the more they spread, causing the images of close objects to overlap.

Q: How do we give electrons a wavelength?

According to quantum mechanics, all matter exhibits wave-like properties. An electron's de-Broglie wavelength is given by $\lambda = \frac{h}{p}$. By accelerating them, we increase their momentum $p$, which drastically decreases their wavelength.

Q: Can we just use X-rays instead of electrons since they also have small wavelengths?

X-rays do have small wavelengths, but it is extremely difficult to focus them because they tend to pass through most materials or get absorbed, rather than refracting through lenses like light or being deflected by magnetic fields like electrons.

Q: Is there any limit to an electron microscope's resolving power?

Yes. While the theoretical limit is incredibly small, practical limits arise from imperfections in the magnetic lenses (spherical and chromatic aberrations) used to focus the electron beam.

Question

Given below are two statements:
One is labelled as Assertion A and the other is labelled as Reason R.
**Assertion A** An electron microscope can achieve better resolving power than an optical microscope.
**Reason R** The de-Broglie's wavelength of the electrons emitted from an electron gun is much less than wavelength of visible light.
In the light of the above statements, choose the correct answer from the options given below.

Options:

(A) A is true but R is false.

(B) Both A and R are true and R is the correct explanation of A.

(C) Both A and R are true but R is not the correct explanation of A.

(D) A is false but R is true.

Sigma Percentile · Accepted Answer

### The Ultimate Limit of Vision

Have you ever wondered why we cannot just keep adding more and more powerful lenses to a standard microscope to see atoms? It feels intuitive—just magnify it more! But physics has a strict speed limit on vision, and it is dictated by the very nature of light itself.

### The Barrier of Light

When we look through an optical microscope, we are using visible light to illuminate the sample. Visible light is a wave, and like all waves, it experiences **diffraction**. When light waves pass through the tiny aperture of a microscope lens, they spread out.

Because of this spreading, if two tiny objects are too close together, their diffracted light waves overlap and blur into a single blob. The ability of a microscope to distinguish these two close objects as separate entities is called its **resolving power**.

Mathematically, the resolving power is given by:
$$ 	ext{RP} = \frac{2\mu \sin	heta}{\lambda} $$

Notice the denominator. The resolving power is **inversely proportional** to the wavelength ($\lambda$) of the light used. Visible light has a wavelength ranging from $400 	ext{ nm}$ to $700 	ext{ nm}$. This places a hard, unbreakable physical limit on how small we can see. Anything smaller than about half the wavelength of light simply cannot be resolved, no matter how perfect your lenses are.

### De-Broglie's Rescue

If the wavelength of light is the problem, the solution is simple: use something with a smaller wavelength! But what?

Enter Louis de-Broglie. He proposed that matter itself has wave-like properties. An electron, which we usually think of as a tiny solid particle, actually behaves like a wave when it moves. The wavelength of this "matter wave" is given by:
$$ \lambda = \frac{h}{p} $$

By accelerating electrons through a high voltage, we give them a massive amount of momentum ($p$). Because momentum is in the denominator, a huge momentum results in a incredibly tiny wavelength. In a typical electron microscope, the de-Broglie wavelength of the electrons is around $0.01 	ext{ nm}$—thousands of times smaller than visible light!

### The Verdict

Now, let's bring it all together and look at our assertion and reason.

**Assertion A** claims that an electron microscope can achieve better resolving power than an optical microscope. Based on our understanding of the resolving power formula, this is absolutely true.

**Reason R** states that the de-Broglie wavelength of electrons is much less than the wavelength of visible light. This is also a factual statement.

More importantly, because $	ext{RP} \propto \frac{1}{\lambda}$, the fact that $\lambda_e \ll \lambda_v$ is the *exact mathematical reason* why the electron microscope has a superior resolving power. Therefore, both statements are true, and the reason perfectly explains the assertion.

Select Answer:

Visualized Solution

Resolving Power of a Microscope

Wavelength: $λ_{v}$ vs $λ_{e}$

Comparing Resolving Powers

Evaluating the Statements

The Sigma Insight: Matter Waves and de Broglie Relation

The Ultimate Limit of Vision

The Barrier of Light

De-Broglie's Rescue

The Verdict

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