Modern Physics formulas for JEE Main & Advanced

23 formula entries for Modern Physics, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Photoelectric Effect

E = h ν = ϕ + K_{max}

When to use. When analyzing light-induced electron emission, calculating stopping potential, or threshold frequency.

Trap. Confusing intensity with frequency: intensity affects number of photoelectrons, not their maximum kinetic energy.

Einstein's Photoelectric Equation

h ν = ϕ + \frac{1}{2} m v_{max}^{2}

When to use. To relate photon energy, work function, and electron kinetic energy in photoelectric problems.

Trap. Using $ν$ (frequency) instead of $f$ ; both are same but be consistent.

Work Function and Threshold Frequency

ϕ = h ν_{0} = \frac{h c}{λ_{0}}

When to use. To determine if photoelectric emission occurs for a given wavelength/frequency.

Trap. Assuming work function is same for all metals; it varies with material.

Stopping Potential Calculation

V_{0} = \frac{h ν - ϕ}{e}

When to use. When given stopping potential data to find $h$ , $ϕ$ , or $K_{max}$ .

Trap. Using $V_{0}$ as kinetic energy directly; it's potential difference, so multiply by $e$ .

de Broglie Wavelength

λ = \frac{h}{p} = \frac{h}{m v}

When to use. When calculating wavelength of electrons, protons, or other particles in motion.

Trap. Applying to photons: $p = h / λ$ works, but $m = 0$ so $λ = h / p$ is consistent.

de Broglie Wavelength for Accelerated Electron

λ = \frac{h}{\sqrt{2 m e V}}

When to use. When an electron is accelerated by a voltage, e.g., in electron microscopes or diffraction experiments.

Trap. Using $m$ in kg, $e$ in C, $V$ in volts; result in meters.

Bohr's Model of Hydrogen Atom

L = m v r = n ℏ

E_{n} = - \frac{13.6}{n^{2}} eV

When to use. For hydrogen-like atoms (single electron) to find energy levels, radii, or transition wavelengths.

Trap. Assuming $n$ starts from 0; it starts from 1.

Rydberg Formula for Hydrogen Spectrum

\frac{1}{λ} = R_{H} (\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}})

When to use. To find wavelengths of spectral lines in hydrogen (Lyman, Balmer, Paschen series).

Trap. Using $n_{i}$ and $n_{f}$ in wrong order; for emission $n_{i} > n_{f}$ .

Nuclear Binding Energy

Δ m = [Z m_{p} + (A - Z) m_{n}] - M_{nucleus}

B . E . = Δ m c^{2}

When to use. To compare stability of nuclei; higher binding energy per nucleon means more stable.

Trap. Using atomic masses instead of nuclear masses; include electron masses carefully.

Radioactive Decay Law

N = N_{0} e^{- λ t}

λ = \frac{\ln 2}{T_{1 / 2}}

When to use. To find remaining nuclei after time $t$ , or half-life from decay data.

Trap. Using $λ$ as decay constant; it's probability per unit time, not half-life.

Solving Decay Problems with Half-Life

N = N_{0} {(\frac{1}{2})}^{t / T_{1 / 2}}

When to use. Quick calculation when time is multiple of half-life; otherwise use $N = N_{0} e^{- λ t}$ .

Trap. Assuming $t / T_{1 / 2}$ must be integer; formula works for any real number.

Mass-Energy Equivalence

E = m c^{2}

When to use. In nuclear reactions, to find energy released from mass defect.

Trap. Using $c = 3 \times 10^{8} m/s$ ; energy in Joules if mass in kg.

Energy of a Photon

E = h ν = \frac{h c}{λ}

When to use. To find energy of a photon given frequency or wavelength.

Trap. Using $λ$ in nm but $h$ in J s; convert to meters.

Compton Effect

Δ λ = λ^{'} - λ = \frac{h}{m_{e} c} (1 - \cos θ)

When to use. When X-rays or gamma rays scatter off electrons, causing wavelength change.

Trap. Assuming shift depends on incident wavelength; it depends only on scattering angle.

Identifying Spectral Series in Hydrogen

Lyman:

n_{i} \geq 2 \to n_{f} = 1

; Balmer:

n_{i} \geq 3 \to n_{f} = 2

; Paschen:

n_{i} \geq 4 \to n_{f} = 3

When to use. To classify spectral lines of hydrogen based on lower energy level.

Trap. Thinking Balmer series is all visible; only first few lines are visible (H-alpha red).

Nuclear Fission and Fusion

Fission:

_{92}^{235} U + n \to fission fragments + 2 - 3 n + energy

When to use. In nuclear energy problems, chain reactions, or stellar energy.

Trap. Assuming fission always produces same products; it's a distribution.

Bohr Radius and Energy for Hydrogen-like Atoms

r_{n} = \frac{n^{2} a_{0}}{Z}

E_{n} = - \frac{13.6 Z^{2}}{n^{2}} eV

When to use. For ions like He $^{+}$ , Li $^{2 +}$ to find energy levels or radii.

Trap. Applying to multi-electron atoms; only valid for single electron systems (H, He $^{+}$ , Li $^{2 +}$ ).

Using Activity and Half-Life in Decay Problems

A = A_{0} e^{- λ t}

λ = \frac{\ln 2}{T_{1 / 2}}

When to use. To find activity after time $t$ , or to determine half-life from activity measurements.

Trap. Forgetting that activity is proportional to number of nuclei.

Photoelectric Effect: Saturation Current

I_{sat} \propto intensity

When to use. When analyzing how photocurrent varies with intensity or applied voltage.

Trap. Assuming saturation current depends on frequency; it depends on number of photons.

Momentum of a Photon

p = \frac{h}{λ} = \frac{E}{c}

When to use. In radiation pressure problems, or Compton effect calculations.

Trap. Using $p = m v$ for photons; photons have no mass, so use $p = E / c$ .

Solving for Wavelength in Hydrogen Transitions

\frac{1}{λ_{limit}} = R_{H} (\frac{1}{n_{f}^{2}})

When to use. To find the boundary wavelength of a spectral series.

Trap. Forgetting that $n_{i} \to \infty$ gives the minimum wavelength in that series.

Mean Life of Radioactive Nucleus

τ = \frac{1}{λ} = \frac{T_{1 / 2}}{\ln 2}

When to use. When given mean life to find decay constant or half-life.

Trap. Confusing mean life with half-life; mean life is about 1.44 times half-life.

Energy Released in Nuclear Reactions

Q = (m_{initial} - m_{final}) c^{2}

When to use. To find energy released or absorbed in any nuclear reaction.

Trap. Using atomic masses without accounting for electrons; for reactions with beta decay, be careful.

Values to remember

$h = 6.63 \times 10^{- 34} J s$ Photoelectric Effect
$e = 1.6 \times 10^{- 19} C$ Photoelectric Effect
$h c = 1240 eV nm$ Einstein's Photoelectric Equation
$1 eV = 1.6 \times 10^{- 19} J$ Einstein's Photoelectric Equation
$ϕ$ typical range: 2-5 eV for metals Work Function and Threshold Frequency
$m_{e} = 9.1 \times 10^{- 31} kg$ de Broglie Wavelength
$λ \approx \frac{1.226}{\sqrt{V}} nm$ for electrons (V in volts) de Broglie Wavelength for Accelerated Electron
$a_{0} = 0.529 \overset{˚}{A}$ Bohr's Model of Hydrogen Atom
$E_{1} = - 13.6 eV$ Bohr's Model of Hydrogen Atom
$R_{H} = 1.097 \times 10^{7} m^{- 1}$ Rydberg Formula for Hydrogen Spectrum
$\frac{1}{λ} = R_{H} Z^{2} (\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}})$ for hydrogen-like Rydberg Formula for Hydrogen Spectrum
$1 amu = 931.5 MeV / c^{2}$ Nuclear Binding Energy
$m_{p} = 1.007276 amu$ Nuclear Binding Energy
$m_{n} = 1.008665 amu$ Nuclear Binding Energy
$\ln 2 \approx 0.693$ Radioactive Decay Law
$c = 3 \times 10^{8} m/s$ Mass-Energy Equivalence
$h = 4.14 \times 10^{- 15} eV s$ Energy of a Photon
$λ_{C} = \frac{h}{m_{e} c} = 0.0243 \overset{˚}{A}$ Compton Effect
Lyman limit: $λ = 91.2 nm$ Identifying Spectral Series in Hydrogen
Balmer limit: $λ = 364.6 nm$ Identifying Spectral Series in Hydrogen
Fission of 1 U-235 releases ~200 MeV Nuclear Fission and Fusion
Fusion of 4 H to He releases ~26.7 MeV Nuclear Fission and Fusion
$E_{1} (H) = - 13.6 eV$ Bohr Radius and Energy for Hydrogen-like Atoms
Lyman limit: 91.2 nm Solving for Wavelength in Hydrogen Transitions
Balmer limit: 364.6 nm Solving for Wavelength in Hydrogen Transitions
$τ = 1.44 T_{1 / 2}$ Mean Life of Radioactive Nucleus

Read the full chapter: Modern Physics for JEE, with what JEE asks and the common traps.