JEE Physics: Modern Physics Complete Guide
Modern Physics is a crucial section in JEE Physics, encompassing the study of atomic and nuclear phenomena, quantum
mechanics, photoelectric effect, and much more. This guide provides an in-depth overview of the topics, formulas,
and concepts needed to excel in this area. Understanding these topics will not only help you score well but also build
a strong foundation for higher studies in physics and engineering.
1. Atomic Models and Structure
Atomic models have evolved to explain the structure and behavior of atoms:
- Thomson’s Model: Also called the “plum pudding” model, where electrons are embedded in a positive sphere.
- Rutherford Model: Shows a tiny dense nucleus with electrons orbiting around it; explained alpha scattering experiment.
- Bohr’s Model: Electrons orbit nucleus in fixed energy levels or shells with quantized angular momentum.
Bohr’s postulates:
- Electrons move in certain allowed circular orbits without radiating energy.
- Angular momentum of electron is quantized:
$$mvr = n\hbar = n\frac{h}{2\pi}$$
where \(n=1,2,3,...\)
- Energy is emitted or absorbed when electron jumps between orbits, given by:
$$\Delta E = h\nu$$
Bohr Radius
The radius of the first orbit in hydrogen atom:
$$a_0 = \frac{4 \pi \epsilon_0 \hbar^2}{m e^2} \approx 0.529 \times 10^{-10} \text{m}$$
Energy Levels in Hydrogen Atom
Energy of \(n^{th}\) orbit is:
$$E_n = -\frac{13.6 \text{ eV}}{n^2}$$
Negative sign indicates bound state. Ionization energy corresponds to \(n \to \infty\).
2. Photoelectric Effect
The photoelectric effect demonstrates the particle nature of light where electrons are ejected from a metal surface
when light of sufficient frequency shines on it.
Einstein’s Photoelectric Equation
The maximum kinetic energy of emitted electrons:
$$K_{max} = h\nu - \phi$$
Where:
\(h\) = Planck's constant \(6.626 \times 10^{-34} \text{Js}\)
\(\nu\) = frequency of incident light
\(\phi\) = work function of metal (minimum energy to eject electron)
Important Points
- Threshold frequency: \( \nu_0 = \frac{\phi}{h} \), below which no electron is emitted.
- Stopping potential (\(V_0\)) relates to max kinetic energy by \(eV_0 = K_{max}\).
- Intensity affects number of electrons but not their kinetic energy.
3. Dual Nature of Matter
Electrons and other particles exhibit wave-particle duality.
De Broglie Wavelength
All matter has an associated wavelength given by:
$$\lambda = \frac{h}{p} = \frac{h}{mv}$$
This was experimentally confirmed by electron diffraction experiments.
4. X-Rays
X-rays are electromagnetic waves with high energy, produced by the deceleration of high-speed electrons (bremsstrahlung)
or characteristic X-rays from atomic transitions.
Duane-Hunt Law
The shortest wavelength in the X-ray spectrum:
$$\lambda_{min} = \frac{hc}{eV}$$
Where \(V\) is the accelerating voltage.
5. Nuclear Physics
Deals with the structure, stability, and reactions of atomic nuclei.
Basic Terms
- Atomic number (\(Z\)): Number of protons.
- Mass number (\(A\)): Number of protons + neutrons.
- Isotopes: Same \(Z\), different \(A\).
- Isobars: Same \(A\), different \(Z\).
- Isotones: Same number of neutrons.
Mass-Energy Relation and Binding Energy
According to Einstein’s relation:
$$E = mc^2$$
The mass defect is the difference between mass of individual nucleons and nucleus. Binding energy:
$$BE = \text{(Mass defect)} \times c^2$$
Binding energy per nucleon indicates stability.
Radioactivity
Spontaneous nuclear decay emits alpha, beta, or gamma radiation.
Radioactive decay law:
$$N = N_0 e^{-\lambda t}$$
Where \(N_0\) is initial nuclei, \(\lambda\) is decay constant.
Half-life \(T_{1/2}\) is related to \(\lambda\):
$$T_{1/2} = \frac{\ln 2}{\lambda}$$
Types of Radioactive Decay
- Alpha decay: Emission of He nucleus (\(^4_2He\)).
- Beta decay: Electron (\(\beta^-\)) or positron (\(\beta^+\)) emission.
- Gamma decay: Emission of high-energy photons.
6. Nuclear Fission and Fusion
Nuclear Fission
Heavy nucleus splits into lighter nuclei releasing large energy, used in nuclear reactors.
Example:
$$^{235}U + n \to ^{92}Kr + ^{141}Ba + 3n + \text{energy}$$
Nuclear Fusion
Light nuclei combine to form heavier nuclei releasing energy, powering stars including the sun.
Example:
$$^2H + ^3H \to ^4He + n + \text{energy}$$
7. Important Constants and Formulas
| Constant |
Value |
Remarks |
| Planck’s constant, \(h\) |
\(6.626 \times 10^{-34} \, \text{Js}\) |
Quantum of action |
| Speed of light, \(c\) |
\(3 \times 10^{8} \, \text{m/s}\) |
Electromagnetic wave speed |
| Electron charge, \(e\) |
\(1.6 \times 10^{-19} \, \text{C}\) |
Elementary charge |
| Mass of electron, \(m_e\) |
\(9.11 \times 10^{-31} \, \text{kg}\) |
Rest mass of electron |
| Bohr radius, \(a_0\) |
\(0.529 \times 10^{-10} \, \text{m}\) |
Radius of hydrogen atom’s 1st orbit |
8. Practice Problems
Problem 1: Calculate the wavelength of an electron moving with velocity \(2 \times 10^6 \, \text{m/s}\).
Solution:
$$\lambda = \frac{h}{mv} = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 2 \times 10^{6}} = 3.64 \times 10^{-10} \text{m}$$
Problem 2: A metal has a work function of 2 eV. Calculate the stopping potential if incident light wavelength is 400 nm.
Solution:
Energy of photon,
$$E = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \times 3 \times 10^{8}}{400 \times 10^{-9}} = 4.97 \times 10^{-19} \text{J} = 3.1 \, eV$$
Maximum kinetic energy:
$$K_{max} = 3.1 - 2 = 1.1 \, eV$$
Stopping potential,
$$V_0 = \frac{K_{max}}{e} = 1.1 \, \text{Volts}$$
9. Tips for JEE Exam
- Understand concepts over rote learning, especially quantum principles.
- Memorize key formulas and constants.
- Practice numerical problems regularly to build accuracy and speed.
- Revise all decay laws and radioactive series.
- Visualize atomic models and transitions to better grasp photoelectric and X-ray concepts.
This comprehensive guide equips you with all the fundamental and advanced concepts in Modern Physics essential for
JEE preparation. Consistent study and problem-solving practice on these topics will ensure a strong performance in the exam.