Animated Solution for Physics - Dual Nature of Matter and Radiation: An electron beam is accelerated by a potential difference $V$ to hit a metallic target to produce X-rays. It produces continuous as well as characteristic X-rays. If ${\lambda }_{\text{min}}$ is the smallest possible wavelength of X-rays in the spectrum, the variation of $\log {\lambda }_{\text{min}}$ with $\log V$ is correctly represented in

Welcome, future physicists! Today, we are going to dive into a fascinating problem from the JEE Main 2017 paper. This isn't just a math problem; it's a window into the quantum world of X-ray production. Imagine an electron, accelerated to incredible speeds, crashing into a metal target. What happens next is pure magic, governed by the laws of quantum mechanics and electromagnetism.

When an electron beam is accelerated by a potential difference $V$, each electron gains a maximum kinetic energy equal to $eV$. As these high-speed electrons strike a metallic target, they undergo rapid deceleration. According to classical electrodynamics, an accelerating (or decelerating) charge emits electromagnetic radiation. In this quantum scenario, the energy lost by the electron is emitted as an X-ray photon.

The most energetic X-ray photon is produced when an electron loses all of its kinetic energy in a single collision. Since the energy of a photon is inversely proportional to its wavelength ($E=\frac{hc}{\lambda }$), this maximum energy corresponds to the minimum possible wavelength, often called the cut-off wavelength or the Duane-Hunt limit.

Mathematically, we equate the maximum kinetic energy to the photon energy:

Rearranging this to solve for the minimum wavelength, we get our master equation:

The question doesn't ask for a simple plot of ${\lambda }_{\text{min}}$ versus $V$. Instead, it asks us to identify the correct graph representing the variation of $\log {\lambda }_{\text{min}}$ with $\log V$. This means we need to translate our physical equation into a logarithmic format.

Let's take the logarithm of both sides of our master equation:

Now, we must deploy our knowledge of logarithm properties. Remember that the logarithm of a quotient is the difference of the logarithms: $\log \left(\frac{A}{B}\right)=\log A-\log B$. We can use this to separate the variables from the constants.

To understand what this graph looks like, let's compare our expanded logarithmic equation to the standard equation of a straight line:

In our case, the variables plotted on the axes are: - The y-axis represents $y=\log ({\lambda }_{\text{min}})$ - The x-axis represents $x=\log V$

By aligning our equation $\log ({\lambda }_{\text{min}})=-\log V+\log \left(\frac{hc}{e}\right)$ with $y=mx+c$, we can immediately identify the key geometric features of the graph:

1. The Slope ($m$): The coefficient of $\log V$ is $-1$. Therefore, the slope is exactly $-1$. A negative slope means the line goes downwards from left to right. 2. The Y-Intercept ($c$): The constant term is $\log \left(\frac{hc}{e}\right)$. Since $h$, $c$, and $e$ are all positive fundamental constants, this term represents a positive y-intercept. This means the line crosses the y-axis above the origin.

Armed with these two pieces of information—a negative slope and a positive y-intercept—we can confidently evaluate the given options.

- Graph (a) shows a positive slope passing through the origin. Incorrect. - Graph (b) shows a positive slope with a positive y-intercept. Incorrect. - Graph (c) shows a zero slope (a horizontal line). Incorrect. - Graph (d) shows a straight line with a negative slope starting from a positive y-intercept.

Graph (d) perfectly matches our mathematical derivation. This problem beautifully illustrates how physical laws can be transformed using basic algebra to reveal linear relationships, a technique frequently used in experimental physics to verify theoretical models.