Modern Physics

Topics covered: Rutherford scattering, Bohr model, X-rays, de Broglie waves, nuclear physics, radioactivity, photoelectric effect.

1) Rutherford Scattering Experiment

• Helped understand the structure of the atom.
• Most α-particles pass undeflected or with small deflection.
• Few α-particles are deflected through large angles (> 90°).

2) Bohr Model of Hydrogen Atom

Postulates

• Electrons revolve in circular orbits under Coulomb attraction.
• Electrons emit or absorb energy only during transitions.
• Angular momentum is quantised: L = n·h / 2π

Key Formulae

Radius of nth orbit:
rₙ = (n²h²) / (4π²mₑk e²) = n²a₀

Velocity of electron:
vₙ = (ke²) / (nħ)

Energy of nth orbit:
Eₙ = −13.6 / n²  eV
  

Energy of emitted photon:

hf = Ei − Ef
  

Rydberg Formula:

1/λ = R (1/nf² − 1/ni²)

R = 1.097 × 10⁷ m⁻¹
  

3) X-rays

• Electromagnetic waves with wavelength range: 0.01 – 1 nm
• Continuous X-rays: Produced due to deceleration of electrons.
• Characteristic X-rays: Due to electronic transitions in atoms.

4) de Broglie (Matter) Waves

λ = h / p

p = mv

λ = h / mv

If kinetic energy = E:
λ = h / √(2mE)
  

For electrons accelerated by potential V:

λ (Å) = 12.27 / √V
  

Bohr Quantisation Condition:

L = mvr = nħ
  

5) Atomic Nucleus

Nuclear radius:
R = R₀ A¹ᐟ³

R₀ ≈ 1.2 × 10⁻¹⁵ m
  

6) Radioactivity

• Radioactive decay is random and spontaneous.
• Independent of temperature, pressure, electric and magnetic fields.
• Decay rate ∝ number of undecayed nuclei.

Decay law:
dN/dt = −λN

N = N₀ e^(−λt)
  

Half-life:

T½ = 0.693 / λ
  

Mean life:

τ = 1 / λ
  

7) Photoelectric Effect

Emission of electrons from a metal surface when light of sufficient frequency falls on it.

Einstein’s equation:
hν = φ₀ + Eₖ(max)

Eₖ(max) = ½ mv² = eV₀
  

Important Observations

• Effect of intensity: Increases photoelectric current, KE unchanged.
• Effect of frequency: Increases KE, current unchanged.
• Threshold frequency (ν₀): Minimum frequency required.
• No time lag between incidence and emission.