IIT-JEE MAINS 2027 cheat sheet for Modern Physics! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Modern Physics: Key Formulas & Concepts

Modern Physics for IIT-JEE Main 2027 covers foundational quantum phenomena, atomic structures, and nuclear physics. Mastery over energy transformations, photoelectric emissions, and decay kinetics is essential for high-yield problem solving.

1. Photoelectric Effect & Matter Waves

Einstein’s photoelectric equation relates maximum kinetic energy \(K_{\text{max}}\), incident photon energy \(E = h\nu\), and work function \(\phi\):

\[K_{\text{max}} = e V_s = h\nu - \phi = \frac{hc}{\lambda} - \phi\]

The de Broglie wavelength \(\lambda\) of a particle with momentum \(p\), accelerated through a potential difference \(V\), is:

\[\lambda = \frac{h}{p} = \frac{h}{\sqrt{2m K}} = \frac{h}{\sqrt{2m e V}}\]

2. Bohr’s Atomic Model

For a hydrogen-like atom of atomic number \(Z\):

• Angular Momentum Quantization: \(L = m v r = \frac{n h}{2\pi}\)
• Radius of \(n\)-th Orbit: \(r_n = 0.529 \times \frac{n^2}{Z} \text{ Å}\)
• Velocity in \(n\)-th Orbit: \(v_n = 2.18 \times 10^6 \times \frac{Z}{n} \text{ m/s}\)
• Total Energy: \(E_n = -13.6 \times \frac{Z^2}{n^2} \text{ eV}\)
• Rydberg Formula: \(\frac{1}{\lambda} = R Z^2 ( \frac{1}{n_1^2} - \frac{1}{n_2^2} )\)

3. Nuclear Physics & Radioactivity

Mass defect \(\Delta m\) and binding energy \(E_b\):

\[E_b = \Delta m \times 931.5 \text{ MeV}, \quad \text{where } \Delta m = \bigl[ Z m_p + (A - Z) m_n \bigr] - M_{\text{nucleus}}\]

Radioactive decay obeys the exponential law:

\[N(t) = N_0 e^{-\lambda t}, \quad T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}, \quad \tau = \frac{1}{\lambda}\]```tikz \begin{tikzpicture}[scale=1.0] \draw[->, thick] (-0.5,0) -- (6.5,0) node[right] {Energy $E$}; \draw[dashed] (0, 0.5) -- (6, 0.5) node[right] {$n=1$ (Ground State, $-13.6\text{ eV}$)}; \draw[dashed] (0, 2.0) -- (6, 2.0) node[right] {$n=2$ ($-3.4\text{ eV}$)}; \draw[dashed] (0, 3.0) -- (6, 3.0) node[right] {$n=3$ ($-1.51\text{ eV}$)}; \draw[dashed] (0, 3.6) -- (6, 3.6) node[right] {$n=4$ ($-0.85\text{ eV}$)}; \draw[thick] (0, 4.3) -- (6, 4.3) node[right] {$n=\infty$ ($0\text{ eV}$)}; \draw[->, red, thick] (2, 2.0) -- (2, 0.5) node[midway, left] {Lyman}; \draw[->, red, thick] (1.2, 3.0) -- (1.2, 0.5); \draw[->, blue, thick] (3.8, 3.0) -- (3.8, 2.0) node[midway, right] {Balmer}; \draw[->, blue, thick] (4.6, 3.6) -- (4.6, 2.0); \end{tikzpicture} ```