Math Solutions

Compare the three types of nuclear radiation: alpha particles (stopped by paper), beta particles (stopped by aluminum), and gamma rays (reduced by thick lead). Show their penetration depths, ionizing power, and physical properties.

Show that localizing a particle's position (narrow wave packet) spreads its momentum distribution, and vice versa. Demonstrate the fundamental limit: Δx · Δp ≥ ℏ/2, where ℏ = h/(2π).

Show the Planck radiation curves for objects at 3000 K, 5000 K, and 7000 K. Find the peak wavelength using Wien's displacement law: λ_max = b/T where b = 2.898 × 10⁻³ m·K. Show how the peak shifts and the total power increases with temperature.

A spaceship has a rest length of L₀ = 100 m. When traveling at v = 0.9c relative to a stationary observer, its observed length contracts to L = L₀/γ = L₀√(1 − v²/c²). Calculate the contracted length and animate the ship shrinking as speed increases.

A spaceship travels at v = 0.9c relative to Earth. Show that 1 second of ship time corresponds to γ = 1/√(1 − v²/c²) = 2.294 seconds of Earth time. Animate two clocks — one on the ship and one on Earth — running at different rates.

Two deuterium nuclei (²H) fuse to form helium-3 plus a neutron, or tritium plus a proton. Show the mass defect and calculate the energy released using E = Δmc². Compare with the proton-proton chain that powers the sun.

A neutron hits a uranium-235 nucleus, causing it to split into barium-141 and krypton-92, releasing 3 neutrons and approximately 200 MeV of energy. Animate the fission process and show how the released neutrons can trigger a chain reaction.

Start with N₀ = 1000 radioactive atoms. The half-life is t½ = 5 seconds. Show the exponential decay N(t) = N₀(1/2)^(t/t½) = N₀ e^(−λt), where λ = ln(2)/t½ is the decay constant. Animate individual atoms randomly decaying, and overlay the mathematical curve.

Calculate and compare the de Broglie wavelength λ = h/mv of an electron (m = 9.109 × 10⁻³¹ kg), a baseball (m = 0.145 kg), and a car (m = 1500 kg) at various speeds. Show why quantum wave effects are observable for electrons but negligible for everyday objects.

Fire individual electrons through a double slit apparatus. Each electron lands as a single point (particle behavior), but over thousands of detections, an interference pattern emerges (wave behavior). Show how this demonstrates wave-particle duality — the central mystery of quantum mechanics.

Calculate and display all spectral lines of hydrogen for the Balmer series (n → 2), Lyman series (n → 1), and Paschen series (n → 3). Use the Rydberg formula: 1/λ = R_H(1/n_f² − 1/n_i²) where R_H = 1.097 × 10⁷ m⁻¹. Show the energy level diagram alongside the spectrum.

Show the Bohr model of hydrogen with electron orbits n = 1 through n = 6. When an electron transitions from a higher orbit to a lower one, a photon is emitted. Calculate the photon energy and wavelength for transitions in the Lyman, Balmer, and Paschen series using E_n = −13.6/n² eV.

Light of increasing frequency is shone on a cesium metal surface (work function φ = 2.1 eV). Determine the threshold frequency below which no electrons are ejected. Above the threshold, calculate the maximum kinetic energy of the emitted electrons using Einstein's equation: KE_max = hf − φ. Show why intensity affects the number of electrons but not their energy.

Show how light is guided through an optical fiber using total internal reflection. A glass core (n₁ = 1.50) is surrounded by a cladding (n₂ = 1.48). Calculate the critical angle at the core-cladding boundary and the maximum acceptance angle (numerical aperture) for light entering the fiber end.

Unpolarized light of intensity I₀ passes through a polarizer and then through a second polarizer (analyzer) whose transmission axis makes angle θ with the first. Show that the transmitted intensity obeys Malus's law: I = I₀ cos²θ / 2. Calculate the intensity at θ = 0°, 30°, 45°, 60°, and 90°.

Show how the human eye adjusts its lens shape (accommodation) to focus on objects at different distances, from the far point (infinity) to the near point (~25 cm). Then show how myopia (nearsightedness) and hypermetropia (farsightedness) arise, and how concave and convex corrective lenses fix them.

A glass lens (n = 1.50) has radii of curvature R₁ = +20 cm (convex front surface) and R₂ = −30 cm (concave back surface). Calculate the focal length using the lens maker's equation: 1/f = (n − 1)[1/R₁ − 1/R₂]. Explore how changing the radii and refractive index changes the focal length and lens shape.

A soap bubble (n = 1.33, thickness 300 nm) is illuminated with white light. Determine which visible wavelengths are reinforced (constructive interference) and which are cancelled (destructive). Explain the role of the phase change on reflection and compute the conditions for bright and dark bands.

Monochromatic light of wavelength 500 nm passes through a single narrow slit of width a = 0.05 mm. A screen is 2 m away. Calculate the width of the central maximum, the positions of the first few minima, and sketch the intensity pattern I(θ) = I₀ [sin(β)/β]² where β = (πa sin θ)/λ.

Monochromatic light of wavelength 600 nm passes through two narrow slits separated by 0.1 mm. A screen is placed 1.5 m away. Calculate the positions of the bright and dark fringes, the fringe spacing, and the intensity pattern. Show how the pattern changes with wavelength and slit separation.