Animated Solution for Physics - Dual Nature of Matter and Radiation: The speed of electrons in a scanning electron microscope is $1\times 10^7{\text{ ms}}^{-1}$. If the protons having the same speed are used instead of electrons, then the resolving power of scanning proton microscope will be changed by a factor of

The resolving power of a microscope dictates its ability to distinguish between two closely spaced objects. In the quantum realm, this resolving power is intimately tied to the wave nature of the particles used to illuminate the sample. Let's dive into how switching from electrons to protons affects this crucial property.

Imagine you are trying to look at the tiny details of a virus. If the "probe" you are using is too large, it will just wash over the details. In electron microscopes, the "probe" is the matter wave associated with the moving electrons.

The fundamental principle is that the Resolving Power (RP) is inversely proportional to the wavelength $\lambda$ of the particles. A smaller wavelength means a sharper, more precise probe, leading to a higher resolving power.

To find the wavelength of these particles, we turn to Louis de Broglie's groundbreaking hypothesis. He proposed that any moving particle has an associated wavelength given by: where $h$ is Planck's constant, $m$ is the mass of the particle, and $v$ is its velocity.

If we substitute this expression for wavelength into our resolving power relation, we get a beautiful new perspective: Since Planck's constant $h$ is just a number, we can simplify this to say that the resolving power is directly proportional to the momentum of the particle:

The problem gives us a very specific constraint: both the electrons and the protons are moving with the exact same speed, $v=1\times 10^7{\text{ ms}}^{-1}$.

Because the velocity $v$ is constant for both cases, it drops out of our proportionality. The resolving power now depends entirely on the mass of the particle! This is a profound realization. By simply using a heavier particle at the same speed, we can achieve a higher resolution.

We are asked to find the factor by which the resolving power changes when we switch to protons. This means we need the ratio of the resolving power of the proton microscope to that of the electron microscope:

We know from fundamental physics that a proton is significantly heavier than an electron. Specifically, the mass of a proton is approximately $1837$ times the mass of an electron ($m_p\approx 1837m_e$).

Substituting this mass ratio into our equation:

The resolving power of the scanning proton microscope will be 1837 times greater than that of the electron microscope. This massive leap in resolution highlights why heavier particles like protons or ions are sometimes used in advanced microscopy techniques!