Modern Physics
Complete Formula Sheet & Shortcut Bible · BITSAT 2026
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Photoelectric Effect & Dual Nature
Einstein's Photoelectric Equation
Kₘₐₓ = hν - φ₀
Kₘₐₓ = Eᵢₙₙ - W
Kₘₐₓ = Eᵢₙₙ - W
Max Kinetic Energy = Incident Energy - Work Function.
Work Function & Thresholds
φ₀ = hν₀ = hc/λ₀
ν₀ is threshold frequency, λ₀ is threshold wavelength.
Stopping Potential (V₀)
Kₘₐₓ = eV₀
Relates max kinetic energy to the potential needed to stop electrons.
Photon Momentum
p = E/c = h/λ
Photons have momentum despite having zero rest mass.
de Broglie Wavelength (General)
λ = h/p = h/mv
Applies to all matter, relating wavelength to momentum.
de Broglie Wavelength (Particles)
λ = h/√(2mK) = h/√(2mqV)
For a particle with charge 'q' accelerated by potential 'V'.
BITSAT Speed Hacks
For photoelectric effect graphs, the slope of Kₘₐₓ vs. frequency (ν) is always Planck's constant (h), independent of the metal.
Use the energy shortcut: E(eV) ≈ 12400 / λ(Å) or E(eV) ≈ 1240 / λ(nm). This is a massive time-saver.
Intensity of light only affects the number of photoelectrons (photocurrent), not their maximum kinetic energy.
In radioactivity, after 'n' half-lives, the remaining amount is N = N₀ / 2ⁿ. The fraction decayed is (1 - 1/2ⁿ).
Bohr's Atomic Model (For H-like atoms)
Radius of nᵗʰ Orbit
rₙ = (0.529 Å) × n²/Z
Directly proportional to n² and inversely to Z.
Velocity in nᵗʰ Orbit
vₙ = (2.18 × 10⁶ m/s) × Z/n
Directly proportional to Z and inversely to n.
Energy in nᵗʰ Orbit
Eₙ = (-13.6 eV) × Z²/n²
Total energy is negative, indicating a bound system.
Angular Momentum Quantization
L = mvr = nh/2π
The fundamental postulate of Bohr's model.
Hydrogen Spectrum
Rydberg's Formula
1/λ = RZ² (1/n₁² - 1/n₂²)
R ≈ 1.097 × 10⁷ m⁻¹ is the Rydberg constant.
Lyman Series (UV)
n₁ = 1
n₂ = 2, 3, 4, ...
n₂ = 2, 3, 4, ...
Transitions ending in the ground state.
Balmer Series (Visible)
n₁ = 2
n₂ = 3, 4, 5, ...
n₂ = 3, 4, 5, ...
The only series with lines in the visible spectrum for Hydrogen.
Paschen Series (IR)
n₁ = 3
n₂ = 4, 5, 6, ...
n₂ = 4, 5, 6, ...
Transitions ending in the second excited state.
Number of Spectral Lines
N = (n₂-n₁)(n₂-n₁+1)/2
For de-excitation from n₂ to ground state (n₁=1), N = n(n-1)/2.
Radioactivity
Law of Radioactive Decay
N(t) = N₀e⁻ˡᵗ
N(t) is the number of undecayed nuclei at time t.
Half-Life (T₁/₂)
T₁/₂ = ln(2)/λ ≈ 0.693/λ
Time for half of the radioactive nuclei to decay.
Mean Life (τ)
τ = 1/λ = T₁/₂ / 0.693
Average lifetime of a nucleus. τ > T₁/₂.
Activity (A)
A(t) = λN(t) = A₀e⁻ˡᵗ
Rate of decay. Units: Becquerel (Bq) or Curie (Ci).
Nuclear Physics & Energy
Nuclear Radius
R = R₀A¹/³
R₀ ≈ 1.2 fm. Nuclear density is nearly constant.
Mass Defect (Δm)
Δm = [Z·mₚ + (A-Z)·mₙ] - Mₙᵤ𝒸
Difference between mass of constituents and mass of the nucleus.
Binding Energy (B.E.)
B.E. = Δm · c²
B.E. (MeV) = Δm (amu) × 931.5
B.E. (MeV) = Δm (amu) × 931.5
Energy required to break a nucleus into its constituent nucleons.
Binding Energy per Nucleon
B.E./A
A measure of nuclear stability. Peaks around A=56 (Iron).
Nuclear Reactions: Fission vs. Fusion
| Property | Nuclear Fission | Nuclear Fusion |
|---|---|---|
Definition | A heavy nucleus splits into two or more lighter nuclei. | Two or more light nuclei combine to form a heavier nucleus. |
Reactants | Heavy nuclei (e.g., ²³⁵U, ²³⁹Pu) | Light nuclei (e.g., ¹H, ²H, ³H) |
Energy Release | ~200 MeV per reaction. High, but less than fusion per nucleon. | ~24 MeV per reaction. Extremely high per nucleon. |
Conditions | Can be initiated by slow neutrons; chain reaction is possible. | Requires extremely high temperature (~10⁷ K) and pressure. |