Osm Physics by Ashish

8. Rutherford Scattering 8.1 Introduction From the study of the trajectory of a body moving under the influence of a central force, the dependence of potential energy on distance, or the law of force, can be determined. From the law of conservation of energy, the total energy of a particle moving under a central force is \[ \boxed{ E = \frac{1}{2}m\left(\frac{dr}{dt}\right)^2 + \frac{J^2}{2mr^2} + U(r) } \tag{1} \] where \(J\) is the angular momentum of the particle and is given by \[ \boxed{ J = mr^2\frac{d\theta}{dt} } \] Rearranging equation (1), \[ \boxed{ U(r) = E - \frac{1}{2}m\left(\frac{dr}{dt}\right)^2 - ...

On the basis of astronomical observations, Johannes Kepler discovered the laws of planetary motion. These laws describe the motion of planets around the Sun and are known as Kepler's laws of planetary motion . First Law: Law of Orbits Every planet moves in an elliptical orbit with the Sun situated at one of the foci of the ellipse. Thus, the orbit of a planet is generally an ellipse rather than a perfect circle. The distance of the planet from the Sun therefore changes continuously during its revolution. Second Law: Law of Areas The line joining a planet and the Sun sweeps out equal areas in equal intervals of time. If \(dA\) is the area swept by the radius vector in a time interval \(...

In polar coordinates, the radial equation of motion for a particle moving under a central force is \[ m\left(\frac{d^2r}{dt^2} -r\left(\frac{d\phi}{dt}\right)^2\right)=F(r). \tag{1} \] Since the force is central, the angular momentum of the particle about the centre of force remains constant. If \(J\) denotes the angular momentum, then \[ J=mr^2\frac{d\phi}{dt}. \] Therefore, \[ \frac{d\phi}{dt} =\frac{J}{mr^2}. \tag{2} \] and \[\because \frac{dr}{dt} = \frac{dr}{d\phi}\frac{d\phi}{dt}. \] Using equation (2), \[ \frac{dr}{dt} = \frac{J}{mr^2}\frac{dr}{d\phi}. \tag{3} \] ...

The second law of thermodynamics states that the entropy of an isolated system never decreases; it either stays the same (in reversible processes) or increases (in irreversible processes). It places fundamental limits on the direction of natural processes and on the efficiency of heat engines and refrigerators. Two classical equivalent formulations are the the Kelvin–Planck (Kelvin) and Clausius statement statement. (i) Kelvin–Planck Statement It is not possible to design an engine which works in a cyclic process and converts all the heat extracted from a heat source into work so that the working substance may remain unaffected. In other words, for the continuous production of work, a heat sink is necessary along with the heat source. According to the original statement given by Ke...

A Carnot cycle is a cyclic process consisting of four reversible processes performed in a definite sequence, namely two isothermal processes and two adiabatic processes. Main Parts of a Carnot Engine (i) Heat Source The heat source is a reservoir of effectively infinite heat capacity maintained at a high temperature \(T_1\) K. Its temperature remains constant even when heat is supplied to the working substance. Its upper surface is perfectly conducting. (ii) Mechanical Arrangement and Working Substance A hollow cylinder is used whose walls are perfectly insulating and whose base is perfectly conducting. A frictionless piston made of insulating material is fitted inside the cylinder. An ideal gas is used as the working substance. (iii) Heat Sink The heat sink is a reservoir of effectively infinite heat capacity maintained at a lower temperature \(T_2...

``` Heat Engine, Carnot Cycle and Efficiency of Carnot Engine Heat Engine A heat engine is a device that converts a part of the heat supplied to it into useful mechanical work. A heat engine must contain the following three essential parts: Heat source Mechanical arrangement and working substance Heat sink (i) Heat Source A heat source is a heat reservoir maintained at a high temperature. It has a very large, ideally infinite, heat capacity so that its temperature remains constant even when a large amount of heat is continuously extracted from it. (ii) Mechanical Arrangement and Working Substance To convert heat into mechanical work, a hollow cylinder fitted with a movable piston is used as the mechanical arrangement. The working substance is placed inside the cylinder. The working substance absorbs heat from the heat source and expands, t...

``` (i) Absolute Temperature is Always Positive From the definition of temperature given by Eqs. (12) and (13) of Section 1.4, \[ \beta = \frac{\partial \ln \Omega(E)}{\partial E} = \frac{1}{k_B T} \tag{1} \] where \( \Omega(E) \) represents the number of accessible microstates corresponding to the energy \(E\). For a general system, \( \Omega(E) \) is a rapidly increasing function of energy \(E\). Therefore, \[ \frac{\partial \ln \Omega(E)}{\partial E} > 0. \] Hence, \( \beta \) is positive. Since \[ \beta = \frac{1}{k_B T}, \] and the Boltzmann constant \(k_B\) is positive, the absolute temperature \(T\) is also positive. Therefore, \[ \boxed{T>0} \] for ordinary physical systems. (ii) Magnitude of Absolute Temperature To estimate the magnitude of temperature, the approximate dependence o...