Osm Physics by Ashish

11. Motion of a System with Varying Mass: Rocket . Rocket A rocket employs the principle of jet propulsion . It may be a missile, spacecraft, or other vehicle that obtains thrust from a rocket engine. The exhaust of a rocket engine is produced entirely from propellants carried within the rocket before its launch. The operation of a rocket engine is based on Newton's third law of motion and the law of conservation of linear momentum . A rocket is propelled forward by ejecting exhaust gases backward at a very high velocity. A rocket engine consists essentially of propellant tanks, a combustion chamber, and a nozzle. The propellants may be gaseous, solid, liquid, or a combination of solid and liquid propellants. In the combustion chamber, a chemical reaction takes place between the fu...

Principle of LASER - Physics Notes Consider an atom having two energy levels \(E_m\) and \(E_n\), where \(E_n > E_m\). Let \(N_m\) and \(N_n\) be the number of atoms in the lower and upper energy levels, respectively. \[ E_n-E_m=h\nu \] The rate of absorption of radiation by atoms in the lower energy state is \[ R_{mn}=N_mP_{mn} \] \[ R_{mn}=N_mB_{mn}u(\nu) \] The total rate of emission from the upper energy state is the sum of spontaneous and stimulated emission: \[ R_{nm}=N_nP_{nm} \] \[ R_{nm}=N_n\left[A_{nm}+B_{nm}u(\nu)\right] \] Therefore, the ratio of total emission rate to absorption rate is \[ \frac{R_{nm}}{R_{mn}} = \frac{N_n}{N_m} \left[ 1+\frac{A_{nm}}{B_{nm}u(\nu)} \right] \tag{1} \] For non-degenerate energy levels, Einstein's relation gives \(B_{mn}=B_{nm}\). Therefore, \[ 1>>\frac{A_{nm}}{B_{nm}u(\nu)} \tag{1} \] Thus \[ \frac{R_{nm}}{R_{mn}}\approx\frac{N_n}{N_m}=e^{-h\nu/k_BT} \] Case-I if T=0 \...

Ratio of Spontaneous and Stimulated Transition Probabilities in Thermal Equilibrium The probability per unit time of spontaneous emission is P n m s p = A n m . P_{nm}^{\mathrm{sp}}=A_{nm}. The probability per unit time of stimulated emission is P n m s t = B n m u ( ν ) . P_{nm}^{\mathrm{st}}=B_{nm}u(\nu). Therefore, the ratio of spontaneous to stimulated emission probabilities is P n m s p P n m s t = A n m B n m u ( ν ) (1) \boxed{ \frac{P_{nm}^{\mathrm{sp}}}{P_{nm}^{\mathrm{st}}} = \frac{A_{nm}}{B_{nm}u(\nu)} } \tag{1} In thermal equilibrium, the spectral energy density of blackbody radiation is given by Planck's radiation law: u ( ν ) = A n m B n m 1 e h ν / k B T − 1 . u(\nu)= \frac{8\pi h\nu^3}{c^3} \frac{1}{e^{h\nu/k_BT}-1}. Using this expression in equation (1), where k B k_B  is Boltzmann's constant and T is the absolute temperature. Case I:  h ν ≫ k B T h\nu\gg k_BT When the photon energy is much greater than the thermal energy, e h ν / k B T ≫ 1. Therefore, P n m s...

Average Energy of a Monatomic Ideal Gas Let a molecule be considered as a small system in thermal equilibrium with a heat reservoir at temperature \(T\). According to the canonical ensemble, the probability that the molecule occupies the \(i\)-th energy state having energy \(\varepsilon_i\) is given by the Boltzmann distribution: \div \[ P_i = \frac{e^{-\beta\varepsilon_i}} {\displaystyle\sum_i e^{-\beta\varepsilon_i}} \tag{1} \] where \[ \beta = \frac{1}{k_BT}, \] \(k_B\) is the Boltzmann constant and \(T\) is the absolute temperature. The average energy of a molecule is given by \[ \bar{\varepsilon} = \sum_i P_i\varepsilon_i. \] Substituting Eq. (1), we obtain \[ \bar{\varepsilon} = \frac{\displaystyle\sum_i \varepsilon_i e^{-\beta\varepsilon_i}} {\displaystyle\sum_i e^{-\beta\varepsilon_i}}. \tag{2} \] Partition Function The canonical partition function is de...

Head-On Inelastic Collision of Two Particles (b) Head-On Inelastic Collision of Two Particles Which Stick Together Consider two particles of masses \( m_1 \) and \( m_2 \). The first particle is moving with velocity \( u_1 \), while the second particle is initially at rest. They collide head-on and, after collision, stick together and move with a common velocity \( v \). (i) In the Laboratory Frame of Reference (L-Frame) Let the initial velocity of particle \( m_1 \) be \( u_1 \), and that of particle \( m_2 \) be zero. According to the law of conservation of linear momentum, \[ m_1 u_1 + m_2(0) = (m_1 + m_2)v \] Therefore, \[ \boxed{v = \dfrac{m_1 u_1}{m_1 + m_2}} \tag{1} \] Loss of Kinetic Energy The kinetic energy before collision is \[ K_i = \dfrac{1}{2} m_1 u_1^2 \] and the kinetic energy after collision is \[ K_f = \dfrac{1}{2}(m_1 + m_2)v^2. \] Using Eq...

Consider a particle of mass m₁, moving with a constant velocity u₁, which collides elastically with a stationary particle of mass m₂ in the laboratory frame of reference. After the collision, the particle of mass m₁ moves with velocity v₁, making an angle θ₁ with its initial direction of motion, while the particle of mass m₂ moves with velocity v₂, making an angle θ₂ with the initial direction of motion of the first particle. Let the initial direction of motion of m₁ be along the X-axis, and let the velocities v₁ and v₂ lie in the X–Y plane, as shown in Fig. 1. Fig. 1: Elastic collision in two dimensions in Lab frame   The velocity of the centre of mass in the laboratory frame is V CM = m₁u₁ / (m₁ + m₂)              (1) The initial velocity of particle m₁ in the C.M. frame is u₁′ = u₁ − V_C   (2) Using Eq. (1), u₁′ = m₂u₁ / (m₁ + m₂)          ...

  If heat is transferred from system A to system A′ (the surroundings) and the temperature of A′ remains unchanged, then system A′ is called a heat reservoir. Let system A be in its i-th state with energy Eᵢ. The combined system A*, consisting of A and A′, is completely isolated. Therefore, its total energy E* remains constant. Hence, when the energy of system A is Eᵢ, the energy of the reservoir A′ is E′ = E* − Eᵢ.    (1) According to the principle of a priori probabilities, all accessible microscopic states of an isolated system are equally probable. Therefore, the probability Pᵢ of finding system A in its specific i-th state is directly proportional to the number of accessible states of the reservoir: Pᵢ ∝ Ω′(E′). Thus, Pᵢ = C′Ω′(E′), or, using Eq. (1), Pᵢ = C′Ω′(E* − Eᵢ).    (2) where C′ is a proportionality constant independent of the i-th state. Since system A is very small compared with the heat reservoir A′, Eᵢ ≪ E*, and therefore E′ ≈ E*. The ...