Category: HC Verma Part 2: Modern Physics

Problem Statement Solve the nuclear physics problem: Using Moseley’s law, find the frequency of the $K_\alpha$ X-ray of copper ($Z = 29$). Moseley’s law: $\sqrt{f} = a(Z-b)$ where $a = 4.97\times10^7$ Hz$^{1/2}$, $b = 1$. $f = a^2(Z-b)^2 = [4.97\times10^7]^2\times(Z-1)^2$ Step 1: $Z-1 = 28$ Step 2: $\sqrt{f} = 4.97\times10^7\times28 = 1.392\times10^9$ Given Information Nuclide…

Problem Statement Solve the oscillation/wave problem: Find the Bragg angle for first-order diffraction of X-rays with $\lambda = 0.154$ nm from planes with $d = 0.282$ nm (NaCl crystal). $\sin\theta = n\lambda/(2d)$ Step 1: $$\sin\theta = \frac{\lambda}{2d} = \frac{0.154}{2\times0.282} = \frac{0.154}{0.564} = 0.2730$$ Step 2: $\theta = \arcsin(0.2730) Given Information Mass $m$ and spring constant…

Problem Statement Solve the oscillation/wave problem: X-rays are diffracted from a crystal with interplanar spacing $d = 0.25$ nm. The first-order maximum is at $\theta = 14°$. Find the wavelength. Bragg’s law: $2d\sin\theta = n\lambda$; $n=1$ Step 1: $\sin 14° = 0.2419$ Step 2: $$\lambda = 2d\sin\theta = 2\times0.25\times0.2419 = 0.1210\text{ nm} = 12 Given…

Problem Statement Solve the oscillation/wave problem: What accelerating voltage is needed to produce X-rays of minimum wavelength $\lambda = 0.1$ nm? $V = hc/(e\lambda) = 1240/(\lambda\text{ [nm]})$ V Step 1: $$V = \frac{hc}{e\lambda} = \frac{1240\text{ eV·nm}}{0.1\text{ nm}} = 12400\text{ V} = 12.4\text{ kV}$$ $$\boxed{V = 12.4\text{ kV}}$$ Given Information Mass $m$ and spring constant $k$…

Problem Statement Solve the oscillation/wave problem: An X-ray tube operates at $V = 30$ kV. Find the minimum wavelength of X-rays produced (Duane-Hunt limit). $\lambda_{min} = hc/(eV) = 1240\text{ eV·nm}/(eV)$ Step 1: $$\lambda_{min} = \frac{hc}{eV} = \frac{1240\text{ eV·nm}}{30000\text{ eV}} = 0.04133\text{ nm} = 41.3\text{ pm}$$ $$\boxed{\lambda_{mi Given Information Mass $m$ and spring constant $k$ (or…

Problem Statement For hydrogen in the $n = 3$ state, find the total number of distinct quantum states (including all $l$ and $m_l$ values). 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.…

Problem Statement Which transition in doubly-ionized lithium (Li$^{2+}$, $Z=3$) gives a wavelength closest to the Balmer $H_\alpha$ line (656 nm)? 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…

Problem Statement Find the ratio of the radii $r_3:r_1$ and the ratio of the energies $|E_3|:|E_1|$ in the Bohr model of hydrogen. 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…

Problem Statement Solve the quantum/modern physics problem: Show that the Bohr quantization condition $L = n\hbar$ is equivalent to the de Broglie standing wave condition $2\pi r = n\lambda_{dB}$. $L = mvr$; $\lambda_{dB} = h/(mv)$ Step 1: Start with $2\pi r = n\lambda_{dB} = n\cdot h/(mv)$ Step 2: Rearrange: $mvr = nh/(2\pi) = n\hbar$ Step…

Problem Statement Find the time taken by the electron to complete one orbit in the $n = 2$ state of hydrogen. 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…