In an electron microscope, to properly resolve a very small object, how must the de Broglie wavelength of the electrons compare with the diameter of the object?

Introduction / Context:
Electron microscopes achieve very high resolving power by using electrons instead of visible light to form images of extremely small objects such as viruses, cell organelles, and crystal lattices. The resolving power of any microscope is fundamentally limited by the wavelength of the radiation or particles used to probe the specimen. This question asks how the de Broglie wavelength of electrons should compare to the size of the object in order to resolve it clearly.

Given Data / Assumptions:

• We are dealing with an electron microscope that uses accelerated electrons having a certain de Broglie wavelength.
• The goal is to resolve a small object of a given diameter, meaning to see it distinctly.
• Resolution in wave based imaging systems is related to the wavelength used.
• We assume standard wave optics ideas and de Broglie wave behaviour of electrons.

Concept / Approach:
In microscopy and wave optics, the Rayleigh criterion states that the minimum resolvable distance between two points is of the order of the wavelength used divided by a numerical factor involving the numerical aperture. To resolve details of size d, the wavelength lambda of the probing radiation or particles should be smaller than or at least comparable to d. If the wavelength is much larger than the object, diffraction effects smear out the image, and fine details cannot be distinguished. For electron microscopes, electrons are accelerated to high energies to reduce their de Broglie wavelength to well below the dimensions of the structures being studied, thereby increasing resolution.

Step-by-Step Solution:
Step 1: Recall that the resolving power of a microscope improves as the wavelength of the radiation used decreases. Step 2: Understand that the de Broglie wavelength of an electron is given by lambda = h / p, where h is Planck constant and p is momentum. Increasing electron speed increases momentum and reduces wavelength. Step 3: For an object of diameter d, meaningful resolution requires lambda to be smaller than or at least comparable to d so that diffraction does not blur the object beyond recognition. Step 4: If lambda is much greater than d, the wave cannot resolve features of that size, similar to trying to see tiny details with very long wavelength radio waves. Step 5: Therefore, to resolve an object in an electron microscope, the electron wavelength should be smaller than or comparable to the object diameter, not larger.

Verification / Alternative check:
Consider visible light with wavelengths around 500 nanometres. Optical microscopes using such light cannot resolve objects much smaller than roughly 200 nanometres due to the diffraction limit. Electron microscopes achieve resolutions down to nanometres or even angstroms because electrons accelerated to high voltages can have de Broglie wavelengths of the order of 0.1 nanometres or less, which is much smaller than the typical features examined. This direct correlation between shorter wavelength and higher resolution confirms that the electron wavelength must be small compared to object dimensions for good resolving power.

Why Other Options Are Wrong:
The electron wavelength should be much greater than the diameter of the object: This would severely limit resolution, making it impossible to see fine details of the object. The electron wavelength should be exactly equal to the diameter of the object: There is no strict requirement of exact equality; the key idea is that wavelength should not exceed the feature size by a large factor and is preferably smaller. Resolution does not depend on the electron wavelength at all: This contradicts fundamental wave optics principles, which show that wavelength is a primary factor in resolving power.

Common Pitfalls:
A common error is to ignore wave behaviour and think only in terms of particle impacts, forgetting that electrons behave as waves at microscopic scales. Another mistake is to assume that using any electrons, regardless of their energy and wavelength, will always give better resolution than light. In reality, only high energy electrons with short de Broglie wavelengths provide superior resolving power. To avoid confusion, always connect resolution to the effective wavelength of the probing radiation or particles.

Final Answer:
To resolve a small object in an electron microscope, the electron wavelength should be smaller than or comparable to the diameter of the object so that fine details can be distinguished.