![Q.59 An X-ray tube operates at 30 kV. If one electron converts 10% of its energy into a photon at first collision, then the wavelength of the photon (correct to two decimal places) is _______ Å. [h = 4.14 × 10⁻¹⁵ eV·s, c = 3 × 10⁸ m s⁻¹ and e = 1.6 × 10⁻¹⁹ C]](https://www.letstalkacademy.com/wp-content/uploads/2025/12/slide_59-12.jpg)
Q.59 An X-ray tube operates at 30 kV. If one electron converts 10% of its energy into a photon at first collision, then the wavelength of the photon (correct to two decimal places) is _______ Å.
[h = 4.14 × 10⁻¹⁵ eV·s, c = 3 × 10⁸ m s⁻¹ and e = 1.6 × 10⁻¹⁹ C]
Problem Analysis
Electrons in the X-ray tube gain kinetic energy from acceleration through 30 kV, calculated as E = eV = (1.6×10-19 C) × (30×103 V) = 4.8×10-15 J, or directly 30,000 eV since energy in eV equals accelerating voltage in volts . At first collision, 10% converts to photon energy: Eγ = 0.1 × 30,000 = 3,000 eV .
Wavelength Calculation
Photon energy relates to wavelength by Eγ = hc/λ, so λ = hc/Eγ. Using h = 4.14×10-15 eV·s, c = 3×108 m/s, hc = 1.242×10-6 eV·pm . Thus, λ = 1.242×10-6/3,000 = 4.14×10-10 m = 4.14 Å (1 Å = 10-10 m) .
Step-by-Step Derivation
- Compute electron energy: 30 keV.
- Photon energy: 0.1 × 30 keV = 3 keV.
- hc product: 4.14×10-15 × 3×108 = 1.242×10-6 eV·pm.
- λ in meters: 4.14×10-10 m.
- Convert: 4.14 Å, correct to two decimal places.
No options provided; this is a numerical fill-in for CSIR NET-style exams .
Core Concept: Bremsstrahlung in X-Ray Production
In an X-ray tube, electrons accelerate through 30 kV, gaining 30 keV kinetic energy (E = eV). Partial energy loss (here 10%) during target collision produces photons via bremsstrahlung, unlike minimum wavelength cases using full energy [web:6][web:9]. Photon energy Eγ = 0.1 × 30,000 = 3,000 eV.
Parameters Table
| Parameter | Value | Unit | Role |
|---|---|---|---|
| Accelerating Voltage | 30,000 | V | Electron energy source |
| Electron Energy | 30,000 | eV | Total KE |
| Photon Energy | 3,000 | eV | 10% conversion |
| hc | 1.242×10-6 | eV·pm | Energy-wavelength link |
| Wavelength | 4.14 | Å | Final result |
Common Mistakes and Exam Tips
Avoid confusing with minimum wavelength λmin = hc/eV (would be 0.414 Å for full 30 kV) [web:4][web:5]. Use eV units consistently; given h is in eV·s simplifies to voltage directly. For CSIR NET, verify units (m to Å) and precision.
Applications in Modern Physics
This X-ray tube wavelength 30 kV 10% energy scenario models diagnostic X-rays and material analysis, linking quantum mechanics to practical tech. Practice variations like 70% conversion (as in 40 kV problems) builds mastery.
