Question

Kinetic energy of a proton is equal to energy '$E$' of a photon. Let '$\lambda_1$' be the de-Broglie wavelength of proton and '$\lambda_2$' be the wavelength of photon. If $\frac{\lambda_1}{\lambda_2} \propto E^n$, then the value of '$n$' is

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $2$
• D. $4$

Hint:

To solve scaling problems like this quickly, focus only on the energy variable power tracker: $\lambda_1 \propto \frac{1}{\sqrt{E}} \rightarrow E^{-1/2}$, and $\lambda_2 \propto \frac{1}{E} \rightarrow E^{-1}$. Taking their ratio gives: $$\frac{\lambda_1}{\lambda_2} \propto \frac{E^{-1/2}}{E^{-1}} = E^{-1/2 - (-1)} = E^{1/2}$$ This confirms $n = 1/2$ within seconds without writing out any full physical constants!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question compares a matter wave property (the de-Broglie wavelength $\lambda_1$ of a proton) with an electromagnetic wave property (the wavelength $\lambda_2$ of a photon). Given that the kinetic energy of the proton equals the total energy $E$ of the photon, we need to find the power dependency exponent $n$ for their wavelength ratio.

Step 2: Key Formula or Approach:
1. The de-Broglie wavelength ($\lambda_1$) of a massive particle with mass $m$ and kinetic energy $E$ is given by: $$\lambda_1 = \frac{h}{\sqrt{2mE}}$$ 2. The wavelength ($\lambda_2$) of a photon with energy $E$ is given by Planck's relation: $$E = \frac{hc}{\lambda_2} \implies \lambda_2 = \frac{hc}{E}$$ 3. Take the ratio $\frac{\lambda_1}{\lambda_2}$ and isolate the energy term $E$ to find its exponent.

Step 3: Detailed Explanation:
Let's write out the ratio of the two wavelengths using their energy equations: $$\frac{\lambda_1}{\lambda_2} = \frac{\left(\frac{h}{\sqrt{2mE}}\right)}{\left(\frac{hc}{E}\right)}$$ The Planck constant $h$ cancels out from both the numerator and denominator: $$\frac{\lambda_1}{\lambda_2} = \frac{1}{\sqrt{2mE}} \times \frac{E}{c} = \frac{E}{c\sqrt{2m}\cdot\sqrt{E}}$$ Simplify the energy terms by dividing $E$ by $\sqrt{E}$: $$\frac{\lambda_1}{\lambda_2} = \frac{\sqrt{E}}{c\sqrt{2m}} = \left(\frac{1}{c\sqrt{2m}}\right) \cdot E^{1/2}$$ Since $c$ and $m$ are fixed constants, the entire term in the brackets is a constant multiplier. This establishes the proportional relationship: $$\frac{\lambda_1}{\lambda_2} \propto E^{1/2}$$ Comparing this to the given expression $\frac{\lambda_1}{\lambda_2} \propto E^n$, we find: $$n = \frac{1}{2}$$

Step 4: Final Answer:
The value of the exponent $n$ is $\frac{1}{2}$, which corresponds to option (A).