Dual Nature of Matter and Radiation - Important Formulas Table

 Dual Nature of Matter and Radiation - Important Formulas Table

S.No.	Concept / Quantity	Formula	Key Notes
1	Energy of Photon	$E = h\nu = \frac{hc}{\lambda}$	Planck’s relation
2	Momentum of Photon	$p = \frac{h}{\lambda} = \frac{E}{c}$	-
3	Einstein’s Photoelectric Equation	$h\nu = \phi_0 + K_{\max}$	-
4	Maximum Kinetic Energy	$K_{\max} = h\nu - \phi_0 = eV_0$	Stopping potential
5	Work Function	$\phi_0 = h\nu_0$	-
6	Threshold Frequency	$\nu_0 = \frac{\phi_0}{h}$	-
7	Threshold Wavelength	$\lambda_0 = \frac{hc}{\phi_0}$	-
8	de Broglie Wavelength	$\lambda = \frac{h}{p} = \frac{h}{mv}$	Matter wave
9	de Broglie Wavelength (Accelerated Electron)	$\lambda = \frac{h}{\sqrt{2meV}}$	V in volts
10	de Broglie Wavelength (in terms of KE)	$\lambda = \frac{h}{\sqrt{2mK}}$	-
11	Heisenberg Uncertainty Principle (Position-Momentum)	$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$	-
12	Heisenberg Uncertainty Principle (Energy-Time)	$\Delta E \cdot \Delta t \geq \frac{h}{4\pi}$	-
13	Number of Photons per second	$n = \frac{P}{h\nu}$	P = power
14	Intensity of Light	$I = \frac{nh\nu}{A}$	-
15	Slope of $K_{\max}$ vs $\nu$ graph	Slope = h (Planck’s constant)	-
16	Cut-off Potential	$V_0 = \frac{h\nu - \phi_0}{e}$	-
17	de Broglie Wavelength for Charged Particle	$\lambda = \frac{h}{\sqrt{2mqV}}$	q = charge
18	Ratio of de Broglie wavelengths (e⁻ & p) at same speed	$\frac{\lambda_e}{\lambda_p} = \sqrt{\frac{m_p}{m_e}}$	-
19	Compton Shift	$\Delta\lambda = \frac{h}{m_0c}(1 - \cos\theta)$	Compton wavelength $\frac{h}{m_0c}$
20	Rest Mass Energy	$E_0 = mc^2$	Einstein mass-energy
21	Maximum wavelength for photoelectric effect	$\lambda_{\max} = \lambda_0$	-
22	Velocity of Electron from de Broglie	$v = \frac{h}{m\lambda}$	-
23	Kinetic Energy from de Broglie	$K = \frac{h^2}{2m\lambda^2}$	-
24	Photon Momentum in eV/c	$p = \frac{E}{c} = \frac{h}{\lambda}$	Useful unit
25	hc in eV-Å	$hc = 12400$ eV Å	Very important constant
26	Minimum Uncertainty in Velocity	$\Delta v \geq \frac{h}{4\pi m \Delta x}$	-
27	Davisson-Germer Relation	$2d \sin\theta = n\lambda$	Electron diffraction
28	Bragg’s Law for Electron Waves	$n\lambda = 2d \sin\theta$	-
29	Photoelectric Current	$I \propto$ Intensity (above threshold)	Saturation current
30	de Broglie Wavelength of Thermal Neutron	$\lambda = \frac{h}{\sqrt{3m kT}}$	-
31	Group Velocity of Matter Wave	$v_g = \frac{d\omega}{dk} = v$ (particle velocity)	-
32	Phase Velocity of Matter Wave	$v_p = \frac{\omega}{k} = \frac{c^2}{v}$	> c
33	Number of Photons in Energy E	$N = \frac{E}{h\nu}$	-
34	Radiation Pressure (Absorption)	$P = \frac{I}{c}$	-
35	Radiation Pressure (Reflection)	$P = \frac{2I}{c}$	-
36	Momentum Transferred by Photon (Absorption)	$\Delta p = \frac{h}{\lambda}$	-
37	Momentum Transferred by Photon (Reflection)	$\Delta p = \frac{2h}{\lambda}$	-
38	de Broglie Frequency	$\nu = \frac{E}{h}$	-
39	Wave Number of Photon	$\bar{\nu} = \frac{1}{\lambda}$	-
40	Stopping Potential for Two Frequencies	$e(V_1 - V_2) = h(\nu_1 - \nu_2)$	-
41	Maximum KE in different metals	$K_{\max1} - K_{\max2} = \phi_2 - \phi_1$	-
42	de Broglie Wavelength of Electron in Bohr Orbit	$2\pi r = n\lambda$	-
43	Minimum Energy of Photon for Emission	$E_{\min} = h\nu_0$	-
44	Relation between λ and V (Electron)	$\lambda (\text{in Å}) = \frac{12.27}{\sqrt{V}}$	Practical formula
45	Relation between λ and V (Proton)	$\lambda (\text{in Å}) = \frac{0.286}{\sqrt{V}}$	-
46	Planck’s Constant from Graph	$h =$ slope of $K_{\max}$ vs $\nu$	-
47	Work Function from Graph	$\phi_0 = h\nu_0$ (x-intercept)	-
48	Uncertainty in Position for Electron	$\Delta x \geq \frac{h}{4\pi m \Delta v}$	-
49	Compton Wavelength of Electron	$\lambda_c = \frac{h}{m_e c} = 0.0243$ Å	-
50	de Broglie Wavelength Ratio (Same KE)	$\frac{\lambda_1}{\lambda_2} = \sqrt{\frac{m_2}{m_1}}$	-

Important Constants

• $h = 6.626 \times 10^{-34}$ J s
• $hc = 12400$ eV Å
• $\frac{h}{4\pi} = 5.27 \times 10^{-35}$ J s
• Electron mass $m_e = 9.1 \times 10^{-31}$ kg
• $c = 3 \times 10^8$ m/s

Note: This comprehensive table covers almost all formulas needed for Class 12 Board Exams, JEE Main, JEE Advanced, and NEET.