Important Dual Nature of Matter & Radiation Formulas for JEE and NEET

1. Energy of a photon

• Formula: E = hν
• Explanation: This formula expresses the energy of a photon in terms of its frequency. Here, E is the energy, h is Planck's constant (6.626 × 10^-34 Js), and ν is the frequency of the photon. The energy of a photon is directly proportional to its frequency.

2. Number of photons emitted per second

• Formula: N = P / E
• Explanation: This formula calculates the number of photons emitted per second. Here, P is the power (energy emitted per second), and E is the energy of a single photon. It shows that the number of photons is inversely proportional to the energy of each photon.

3. Momentum of a photon

• Formula: P = E / c = h / λ
• Explanation: The momentum of a photon is calculated using this formula. P is the momentum, E is the energy of the photon, and c is the speed of light. Alternatively, using the relation E = hν, we can express momentum as P = h / λ where λ is the wavelength of the photon.

4. Equivalent mass of a photon

• Formula: m = E / c² = h / λc
• Explanation: A photon, although massless in the classical sense, has equivalent mass derived from its energy. Using Einstein's mass-energy equivalence E = mc², we can derive the mass of a photon in terms of its energy and wavelength.

5. Work function

• Formula: W₀ = hν₀
• Explanation: The work function is the minimum energy required to remove an electron from a material (typically a metal). Here, W₀ is the work function, and ν₀ is the threshold frequency required to emit an electron.

6. Kinetic energy of photoelectron (Einstein's photoelectric equation)

• Formula: K_max = 1/2 mv² = h(ν - ν₀)
• Explanation: This equation describes the maximum kinetic energy (K_max) of the photoelectron emitted when a photon hits a material. m is the mass of the emitted electron, v is its velocity, h is Planck's constant, and ν₀ is the threshold frequency. The kinetic energy is proportional to the difference between the frequency of the incident light and the threshold frequency.

7. Kinetic energy if V₀ is the stopping potential

• Formula: K = 1/2 m v_max² = - eV₀
• Explanation: The stopping potential V₀ is the potential required to stop the most energetic photoelectron. The kinetic energy of the emitted electron is directly related to the stopping potential.

8. Kinetic energy of De-Broglie Waves

• Formula: K = 1/2 mv² = p² / 2m
• Explanation: The De-Broglie wave theory postulates that particles, like electrons, have a wavelength associated with them. The kinetic energy of the particle can be expressed in terms of its momentum (p) and mass m.

9. Momentum of De-Broglie Waves

• Formula: P = √(2mK)
• Explanation: This formula relates the momentum of a particle to its kinetic energy K and mass m. It shows that the momentum is proportional to the square root of the product of mass and kinetic energy.

10. Wavelength of De-Broglie Waves

• Formula: λ = h / p = h / √(2mK)
• Explanation: The wavelength λ associated with a particle's De-Broglie wave is inversely proportional to its momentum P. Here, h is Planck’s constant, and m is the mass of the particle, while K is its kinetic energy.

11. De-Broglie wavelength of an electron beam

• Formula: λ = h / √(2meV)
• Explanation: The De-Broglie wavelength of an electron accelerated through a potential difference V is given by this formula. e is the charge of the electron, and m is its mass. This formula shows that higher the potential difference, the smaller the De-Broglie wavelength.

12. De-Broglie wavelength associated with gas molecules

• Formula: λ = h / √(2mkT)
• Explanation: This formula describes the De-Broglie wavelength of gas molecules, where m is the mass of the molecules, k is Boltzmann’s constant, and T is the temperature in Kelvin. It shows the relationship between the wavelength and the temperature of the gas.

13. The value of h/c in eV ⋅ m

• Formula: h / c = 1240 × 10^-9 eV ⋅ m
• Explanation: This value is a constant used for converting photon energy from wavelength to eV. It is a useful conversion factor when dealing with electromagnetic waves.

14. The value of h/c in eV ⋅ m

• Formula: h / c = 1240 × 10^-9 eV ⋅ m
• Explanation: This is another form of the constant used in calculating energy of a photon with given wavelength in eV. The same value as above is used in solving problems involving photon energy.