Section 10.8: Review Examples
These examples cover topics from Atomic Models, Quantum Theory, Photoelectric Effect, Wave-Particle Duality, Nuclear Physics, Radioactivity, and Special Relativity.
Example 1: Energy of a Photon
A photon has a wavelength of 500 nm. Find its energy.
Energy: \( E = \frac{hc}{\lambda} = \frac{6.626\times10^{-34} \times 3\times10^8}{500\times10^{-9}} \approx 3.97\times10^{-19} \text{ J} \)
Example 2: Half-Life Problem
A radioactive isotope has a half-life of 3 hours. How much of a 10 g sample remains after 9 hours?
\( N = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}} = 10 \left(\frac{1}{2}\right)^{9/3} = 10 \times \frac{1}{8} = 1.25 \text{ g} \)
Example 3: Time Dilation
A spaceship travels at 0.6c. If 4 hours pass on the ship, how much time passes on Earth?
\( t = \frac{t_0}{\sqrt{1 - v^2/c^2}} = \frac{4}{\sqrt{1 - 0.6^2}} \approx 5 \, \text{hours} \)
Practice Problems
- Calculate the energy of a photon of wavelength 400 nm.
- An isotope has a half-life of 10 days. How much of a 50 g sample remains after 30 days?
- Compute the relativistic momentum of a 1 kg particle moving at 0.8c.
- Find the contracted length of a rod of proper length 5 m moving at 0.7c.
- Explain why photons have zero rest mass but non-zero momentum.