• Rutherford’s picture of the atom (a tiny positive nucleus surrounded by orbiting electrons) had a serious problem.
• According to classical electromagnetism, an electron moving round the nucleus is accelerating, so it should continuously radiate energy, spiral into the nucleus in a tiny fraction of a second, and emit light of continuously changing frequency.
• Real atoms are stable, and they emit light only at certain definite frequencies (line spectra).
• Bohr resolved the difficulty by adding quantum ideas to the nuclear atom.

• The lines are grouped into series, according to the level to which the electron falls (the final level n_f):

• Within each series the first line (called α) comes from n_i = n_f + 1 and has the longest wavelength of that series; the lines get closer together and approach the series limit (from n_i = ∞), which is the shortest wavelength of the series.
• Typical values:

1. We can classify the spectra in two ways.
2. By distribution:

• The Coolidge tube is a high-vacuum glass tube containing:

• a hot filament (cathode) heated by a low-voltage current, which emits electrons by thermionic emission;
• a target (anode), made of a metal with a high melting point such as tungsten, set at a slope in front of the cathode;
• a high DC voltage (tens of kilovolts) between filament and target, which accelerates the electrons so that they hit the target with large kinetic energy (eV);
• a copper rod with cooling fins, which removes the heat produced (most of the electrons’ energy, over 99 %, is turned to heat; a tiny part becomes X-rays).
• When an electron collides with the tungsten target, part or all of its kinetic energy is converted to X-ray photons.

• Take an electron (mass m, charge e) moving in a circle of radius r with speed v round a nucleus of charge +e.
• Step 1 — force balance. The Coulomb attraction provides the centripetal force:
• K e²/r² = m v²/r, where K = 9 × 10⁹ N·m²/C². Hence m v² = K e²/r. (i)
• Step 2 — quantum (standing-wave) condition. Only orbits whose circumference holds a whole number of de Broglie wavelengths are allowed: 2π r = n λ = n h/(m v), so m v r = n h/2π. (ii)
• Step 3 — radius. From (ii), v = n h/(2π m r). Substitute into (i): m n² h²/(4π² m² r²) = K e²/r, which gives
• r_n = n² h²/(4π² m K e²) = n² r₁, with r₁ = 5.3 × 10⁻¹¹ m = 0.053 nm.
• Step 4 — energy. Kinetic energy = ½mv² = K e²/(2r). Potential energy = −K e²/r. The total is
• E_n = −K e²/(2 r_n) = −(K e²/2 r₁)/n² = −13.6/n² eV.
• (You do not need to reproduce the derivation in the exam, but you should understand what it shows: the energy is quantized because the radius is quantized, and the radius is quantized because the electron wave must fit the orbit.)

• 📖

🔍 In-Depth Explanations, Deductions & Multi-Book Insights:

Physics (Thanaweya Amma 2026/2027)

• Lesson 1 – Hydrogen Atomic Emission Spectrum and Production of X-Rays
• The hydrogen spectrum provided experimental verification for Niels Bohr's atomic model.
• When an excited electron transitions to a lower energy level, a photon of energy ΔE = Eupper – Elower = h·ν is emitted across distinct spectral series (Lyman, Balmer, Paschen, Brackett, Pfund).
• X-rays are produced in Coolidge tubes via continuous Bremsstrahlung and characteristic line emissions.

🌳 Conceptual Hierarchy Tree

Atomic Spectra and X-Ray Physics

├── 1. Hydrogen Spectral Series

└── 2. X-Ray Production (Coolidge Tube)

• ├── Continuous Spectrum (Bremsstrahlung / Braking Radiation):
• │ ├── Electrons decelerate near heavy target nuclei; λ_min = (h · c) / (e · V)
• │ └── Depends ONLY on the accelerating potential difference V
• └── Characteristic Line Spectrum:
• ├── Incident electron knocks out inner-shell (K or L) electron; outer electron falls into vacancy
• └── Depends ONLY on the Atomic Number (Z) of target material (Tungsten)

📋 Master Teacher's Blackboard Secrets