Modern Physics Formula Sheet: The Last-Minute Revision Guide for JEE
All essential formulas, constants, and definitions from Modern Physics unit - perfectly organized for quick revision.
Essential Physical Constants
$c = 3 \times 10^8 \text{m/s}$
$h = 6.63 \times 10^{-34} \text{J·s}$
$e = 1.6 \times 10^{-19} \text{C}$
$m_e = 9.1 \times 10^{-31} \text{kg}$
$m_p = 1.67 \times 10^{-27} \text{kg}$
$k = 1.38 \times 10^{-23} \text{J/K}$
$R = 1.097 \times 10^7 \text{m}^{-1}$
$N_A = 6.022 \times 10^{23} \text{mol}^{-1}$
$1 \text{u} = 931.5 \text{MeV/c}^2$
1. Photoelectric Effect
Energy of Photon
$$E = h\nu = \frac{hc}{\lambda}$$
where $\nu$ = frequency, $\lambda$ = wavelength
Einstein's Photoelectric Equation
$$h\nu = \phi + K_{max}$$
$\phi$ = work function, $K_{max}$ = maximum kinetic energy
Cut-off Frequency
$$\nu_0 = \frac{\phi}{h}$$
Minimum frequency for photoemission
Stopping Potential
$$eV_0 = K_{max} = h\nu - \phi$$
$V_0$ = stopping potential
Maximum Kinetic Energy
$$K_{max} = h(\nu - \nu_0)$$
When $\nu > \nu_0$
Photocurrent
$$I \propto \text{Intensity}$$
Independent of frequency
🎯 Key Points to Remember
- Photoelectric effect demonstrates particle nature of light
- Maximum KE depends on frequency, not intensity
- Photocurrent depends on intensity, not frequency
- No time lag between illumination and emission
2. Atomic Physics
Bohr's Postulates
$$mvr = \frac{nh}{2\pi}$$
Angular momentum quantization
Radius of nth Orbit
$$r_n = \frac{n^2h^2}{4\pi^2kZe^2m}$$
For hydrogen: $r_n = 0.529 \text{Å} \times n^2$
Velocity in nth Orbit
$$v_n = \frac{2\pi ke^2}{nh}$$
For hydrogen: $v_n = \frac{2.18 \times 10^6}{n} \text{m/s}$
Energy of nth Orbit
$$E_n = -\frac{2\pi^2k^2Z^2e^4m}{n^2h^2}$$
For hydrogen: $E_n = -\frac{13.6}{n^2} \text{eV}$
Rydberg Formula
$$\frac{1}{\lambda} = RZ^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$
For hydrogen-like atoms
Photon Energy in Transition
$$\Delta E = E_2 - E_1 = h\nu$$
Energy of emitted/absorbed photon
3. Nuclear Physics
Nuclear Size
$$R = R_0A^{1/3}$$
$R_0 = 1.2 \text{fm}$, A = mass number
Binding Energy
$$BE = [Zm_p + (A-Z)m_n - M]c^2$$
M = mass of nucleus
Binding Energy per Nucleon
$$\frac{BE}{A} = \frac{[Zm_p + (A-Z)m_n - M]c^2}{A}$$
Maximum for iron (Fe-56)
Radioactive Decay Law
$$N = N_0e^{-\lambda t}$$
$\lambda$ = decay constant
Half-life
$$T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$$
Time for half the nuclei to decay
Activity
$$A = \lambda N = A_0e^{-\lambda t}$$
Decays per second
🎯 Nuclear Reactions
Fission:
$^{235}_{92}U + ^1_0n \rightarrow ^{141}_{56}Ba + ^{92}_{36}Kr + 3^1_0n + Q$
Fusion:
$^2_1H + ^3_1H \rightarrow ^4_2He + ^1_0n + 17.6 \text{MeV}$
4. Semiconductor Devices
Electrical Conductivity
$$\sigma = e(n_e\mu_e + n_h\mu_h)$$
$n$ = carrier concentration, $\mu$ = mobility
Resistivity
$$\rho = \frac{1}{\sigma}$$
Inverse of conductivity
Diode Current
$$I = I_0(e^{eV/kT} - 1)$$
$I_0$ = reverse saturation current
Energy Gap
$$E_g = E_c - E_v$$
Conduction band - Valence band
Intrinsic Carrier Concentration
$$n_i^2 = n_e n_h$$
For intrinsic semiconductors
Transistor Current Gain
$$\beta = \frac{I_c}{I_b}$$
Common emitter configuration
5. Wave-Particle Duality & X-rays
de Broglie Wavelength
$$\lambda = \frac{h}{p} = \frac{h}{mv}$$
For massive particles
For accelerated electron
$$\lambda = \frac{12.27}{\sqrt{V}} \text{Å}$$
V in volts
Heisenberg Uncertainty Principle
$$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$$
Position-momentum uncertainty
X-ray Minimum Wavelength
$$\lambda_{min} = \frac{hc}{eV}$$
V = accelerating voltage
Bragg's Law
$$2d\sin\theta = n\lambda$$
d = interplanar spacing
Moseley's Law
$$\sqrt{\nu} = a(Z - b)$$
For characteristic X-rays
🚀 Last-Minute Revision Strategy
High-Yield Formulas
- Photoelectric effect: $K_{max} = h\nu - \phi$
- Bohr model: $E_n = -13.6/n^2 \text{eV}$
- Radioactive decay: $N = N_0e^{-\lambda t}$
- de Broglie: $\lambda = h/p$
- Stopping potential: $eV_0 = K_{max}$
Common Mistakes to Avoid
- Confusing work function with threshold frequency
- Mixing up fission and fusion reactions
- Forgetting units conversion (eV to Joules)
- Not remembering constants (h, c, e, m_e)
- Confusing n-type and p-type semiconductors
🧠 Memory Aids & Mnemonics
Photoelectric Effect
"Keep Every Photon For Work"
$K_{max} = E_{photon} - \phi_{work}$
Radioactive Decay
"Never Expect Long Time"
$N = N_0e^{-\lambda t}$
Bohr's Model
"Radius Varies with n², Energy with 1/n²"
$r_n \propto n^2$, $E_n \propto 1/n^2$
de Broglie
"Long Wavelength for Slow Particles"
$\lambda = h/p$
You're Ready for Modern Physics!
You now have all the essential formulas at your fingertips. Practice applying them!