A particle having electric charge 3 times 10^-19 text C$3 \times 10^{-19} \text{ C}$ and mass 6 times 10^-27 text kg$6 \times 10^{-27} \text{ kg}$ is accelerated by applying an electric potential of 1.21 text V$1.21 \text{ V}$. Wavelength of the matter wave associated with the particle is alpha times 10^-12 text m$\alpha \times 10^{-12} \text{ m}$. The value of alpha$\alpha$ is ________. (Take Planck's constant = 6.6 times 10^-34 text Jcdottexts$= 6.6 \times 10^{-34} \text{ J}\cdot\text{s}$)

### Related Formula lambda = frachp = frachsqrt2mK$$\lambda = \frac{h}{p} = \frac{h}{\sqrt{2mK}}$$ K = qV$K = qV$ ### Core Logic For a charged particle accelerated through a potential difference V$V$, the de Broglie wavelength is: lambda = frachsqrt2mqV$$\lambda = \frac{h}{\sqrt{2mqV}}$$ ### Step 1: Value Substitution lambda = frac6.6 times 10^-34sqrt2 times 6 times 10^-27 times 3 times 10^-19 times 1.21$$\lambda = \frac{6.6 \times 10^{-34}}{\sqrt{2 \times 6 \times 10^{-27} \times 3 \times 10^{-19} \times 1.21}}$$ lambda = frac6.6 times 10^-34sqrt36 times 10^-46 times 1.21$$\lambda = \frac{6.6 \times 10^{-34}}{\sqrt{36 \times 10^{-46} \times 1.21}}$$ ### Step 2: Arithmetic Evaluation Evaluate the square root: sqrt36 times sqrt1.21 times sqrt10^-46 = 6 times 1.1 times 10^-23 = 6.6 times 10^-23$$\sqrt{36} \times \sqrt{1.21} \times \sqrt{10^{-46}} = 6 \times 1.1 \times 10^{-23} = 6.6 \times 10^{-23}$$ lambda = frac6.6 times 10^-346.6 times 10^-23$$\lambda = \frac{6.6 \times 10^{-34}}{6.6 \times 10^{-23}}$$ lambda = 10^-11 text m$$\lambda = 10^{-11} \text{ m}$$ ### Step 3: Final Conclusion We need to express this in the form alpha times 10^-12 text m$\alpha \times 10^{-12} \text{ m}$: 10^-11 text m = 10 times 10^-12 text m$$10^{-11} \text{ m} = 10 \times 10^{-12} \text{ m}$$ Therefore, alpha = 10$\alpha = 10$. ### Pattern Recognition The expression inside the radical always evaluates cleanly in JEE. 2 times 6 times 3 = 36$2 \times 6 \times 3 = 36$ and 1.21 = (1.1)^2$1.21 = (1.1)^2$ are engineered to perfectly cancel the 6.6$6.6$ in the numerator. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Dual Nature of Radiation and Matter