Category: HC Verma Part 2: Modern Physics

Problem Statement Solve the kinematics problem: Find the speed of the electron in the first Bohr orbit of hydrogen. $v_n = e^2/(4\pi\varepsilon_0 \hbar n) = \alpha c / n$ where $\alpha = 1/137$ $v_1 = \alpha c = c/137$ Step 1: $v_1 = ke^2/(\hbar) = (9\times10^9\times(1.6\times10^{-19})^2)/(1.055\times10^{-34})$ Step 2: Numerator: $9\times10^9\tim Given Information Initial velocity $u$…

Problem Statement Solve the oscillation/wave problem: Find the wavelength of the first line of the Lyman series ($n=2 \to n=1$) in hydrogen. $\frac{1}{\lambda} = R_H(1/1^2 – 1/2^2)$ Step 1: $$\frac{1}{\lambda} = 1.097\times10^7\left(1 – \frac{1}{4}\right) = 1.097\times10^7\times\frac{3}{4} = 8.228\times10^6\text{ m}^{-1}$$ Step 2: $$\lambda = \frac{1}{ Given Information Mass $m$ and spring constant $k$ (or equivalent), or…

Problem Statement Solve the oscillation/wave problem: Find the wavelength of the first line of the Balmer series (transition $n=3 \to n=2$) in hydrogen. $\frac{1}{\lambda} = R_H\left(\frac{1}{2^2} – \frac{1}{3^2}\right)$; $R_H = 1.097\times10^7$ m$^{-1}$ Step 1: $$\frac{1}{\lambda} = 1.097\times10^7\left(\frac{1}{4} – \frac{1}{9}\right) = 1.097\times10 Given Information Mass $m$ and spring constant $k$ (or equivalent), or wave parameters Initial…

Problem Statement Solve the work-energy problem: Find the energy of the hydrogen atom in the $n = 3$ orbit. $E_n = -13.6/n^2$ eV Step 1: $$E_3 = \frac{-13.6}{3^2} = \frac{-13.6}{9} = -1.51\text{ eV}$$ $$\boxed{E_3 = -1.51\text{ eV}}$$ Given Information Mass $m$, velocity $v$, height $h$, or other given quantities Any forces doing work (conservative or…

Problem Statement Find the radius of the first Bohr orbit of hydrogen atom. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) Physical Concepts & Formulas This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation…

Problem Statement Find the minimum frequency of a photon that can ionize a hydrogen atom in its ground state ($E_1 = -13.6$ eV). Also find the corresponding wavelength. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) Physical Concepts & Formulas This problem draws…

Problem Statement Solve the oscillation/wave problem: Solve the quantum/modern physics problem: Find the de Broglie wavelength of a nitrogen molecule ($M = 28$ g/mol) at room temperature $T = 300$ K. Average KE of molecule: $\frac{3}{2}kT = \frac{p^2}{2m}$; so $p = \sqrt{3mkT}$ $\lambda = h/p$ Step 1: Mass of one molecule: $m = 28\times10^{-3}/6.022\ti Given…

Problem Statement Solve the oscillation/wave problem: Solve the quantum/modern physics problem: For a metal, the threshold wavelength for photoelectric emission is $\lambda_0 = 600$ nm. Light of wavelength $\lambda = 400$ nm is incident on it. Find the stopping potential. $\phi = hc/\lambda_0$; $eV_s = hc/\lambda – \phi = hc(1/\lambda – 1/\lambda_0)$ S Given Information…

Problem Statement Solve the quantum/modern physics problem: Solve the quantum/modern physics problem: How many photons per second are emitted by a 60 W bulb, assuming it radiates all energy as visible light of wavelength $\lambda = 550$ nm? $N = P/E_{ph} = P\lambda/(hc)$ Step 1: $E_{ph} = 1240/550 = 2.255$ eV $= 3.61\times10^{-19}$ J Step…

Problem Statement Solve the oscillation/wave problem: Solve the quantum/modern physics problem: Calculate the energy of a photon of wavelength (a) $\lambda = 1$ nm (X-ray), (b) $\lambda = 500$ nm (visible), (c) $\lambda = 10$ cm (microwave). $E = hc/\lambda = 1240\text{ eV·nm}/\lambda[\text{nm}]$ (a) $\lambda = 1$ nm: $E = 1240/1 = 1240$ eV $=…