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Showing posts from January, 2023

Galilean Transformation Equations and Failure of Galilean Relativity

What is Transformation Equation? A point or a particle at any instant, in space has different cartesian coordinates in the different reference systems. The equation which provide the relationship between the cartesian coordinates of two reference system are called Transformation equations. Galilean Transformation Equation: Let us consider, two frames $S$ and $S'$ in which frame $S'$ is moving with constant velocity $v$ relative to an inertial frame $S$. Let The origin of the two frames coincide at $t=0$ The coordinate axes of frame $S'$ are parallel to that of the frame $S$ as shown in the figure below The velocity of the frame $S'$ relative to the frame $S$ is $v$ along x-axis; The position vector of a particle at any instant $t$ is related by the equation $ \overrightarrow{r'}=\overrightarrow{r}-\overrightarrow{v}t\qquad (1)$ In component form, the coordinate are related by the equations $\overrightarrow{

Consequences of Lorentz's Transformation Equations

Consequences: There are two consequences of Lorentz's Transformation Length Contraction (Lorentz-Fitzgerald Contraction) Time Dilation (Apparent Retardation of Clocks) Length Contraction (Lorentz-Fitzgerald Contraction): Lorentz- Fitzgerald, first time, proposed that When a body moves comparable to the velocity of light relative to a stationary observer, then the length of the body decreases along the direction of velocity. This decrease in length in the direction of motion is called ' Length Contraction ' . Expression for Length Contraction: Let us consider two frames $S$ and $S'$ in which frame $S'$ is moving with constant velocity $v$ relative to frame $S$ along the positive x-axis direction. Let a rod is associated with frame $S'$. The rod is at rest in frame $S'$ so the actual length $l_{0}$ is measured by frame $S'$. So $ l_{0}=x'_{2}-x'_{1}\quad\quad (1)$

Derivation of Lorentz Transformation's Equations

Derivation: Let us consider two inertial frames $S$ and $S'$ in which frame $S'$ is moving with constant velocity $v$ along the positive x-axis direction relative to the frame $S$. Let $t$ and $t'$ be the time recorded in two frames. Let the origin $O$ and $O'$ of the two reference systems coincide at $t=t'=0$. Now suppose, a source of light is situated at the origin $O$ in the frame $S$, from which a wavefront of light is emitted at time $t=0$. When the light reaches point $P$, the time required by a light signal in travelling the distance OP in the Frame $S$ is $ t=\frac{OP}{c}$ $ t=\frac{\left (x^{2}+y^{2}+z^{2} \right )}{c}$ $ x^{2}+y^{2}+z^{2}=c^{2}t^{2}\qquad (1)$ The equation $(1)$ represents the equation of wavefront in frame $S$. According to the special theory of relativity, the velocity of light will be $c$ in the second frame $S'$. Hence in frame $S'$ the time required by the light signal in travell

Force between multiple charges (Superposition principle of electrostatic forces)

Principle of Superposition for Electric force: If a system contains a number of interacting charges, then the net force on anyone charge equals the vector sum of all the forces exerted on it by all the other charges. This is the principle of Superposition for electric force . If a system contains n point charges $ q_{1},q_{2},q_{3}........q_{n}$. Then according to the principle of superposition, the force acting on the charge $q_{1}$ due to all the other charges $\overrightarrow{F_{1}}=\overrightarrow{F_{12}}+\overrightarrow{F_{13}}+\overrightarrow{F_{14}}+...+\overrightarrow{F_{1n}} \qquad (1)$ Where $\overrightarrow{F_{12}}$ is the force on charge $q_{1}$ due to charge $q_{2}$, $\overrightarrow{F_{13}}$ that is due to $q_{3}$ and $\overrightarrow{F_{1n}}$ that due to $q_{n}$. If the distance between the charges $q_{1}$ and $q_{2}$ is $\widehat{r}_{12}$ (magnitude only) and $\widehat{r}_{21}$ is unit vector from charge $q_{2}$ to $q_{1}$, then $\over

Definition and derivation of the phase velocity and group velocity of wave

Wave: A wave is defined as a disturbance in a medium from an equilibrium condition that propagates from one region of the medium to other regions. When such type of wave propagates in the medium a progressive change in phase takes place from one particle to the next particle. Propagation of Wave: Wave propagation in the medium occurs with two different kinds of velocity. i.e. phase velocity and group velocity. 1. Phase Velocity: The velocity with which plane of constant the phase of a wave propagates through the medium at a certain frequency is called the phase velocity or wave velocity. A plane wave traveling in the positive x-direction is represented by $y=A \: sin \omega (t-\frac{x}{v})$ Where $ \omega $ – angular frequency $y=A \: sin(\omega t-\frac{\omega x}{v})$ $y=A \: sin(\omega t-kt) \qquad(1)$ For plane-wave $(\omega t-kx)$ is the phase of wave motion. For the plane of constant phase (wavefront). We have $( \omega t-kx) = constan

Product of phase velocity and group velocity is equal to square of speed of light

Prove that $\rightarrow$ The Product of phase velocity and group velocity is equal to the square of the speed of light i.e. $\left( V_{p}.V_{g}=c^{2} \right)$ Proof → We know that $V_{p}=\nu \lambda \qquad(1)$ And de Broglie wavelength- $\lambda =\frac{h }{mv}\qquad(2)$ According to Einstein's mass-energy relation- $E=mc^{2}$ $h\nu=mc^{2}$ $\nu=\frac{mc^{2}}{h}\qquad(3)$ Now put the value of $\lambda $ and $\nu $ in equation$(1)$ $V_{p}= [\frac{mc^{2}}{h}] [\frac{h}{mv}]$ $V_{p}=\frac{C^{2}}{v}$ Since group velocity is equal to particle velocity i.e. $V_{g}=v$. So above equation can be written as $V_{p}=\frac{C^{2}}{V_{g}}$ $V_{p}.V_{g}=C^{2}$ Note → ➢ $V_{g}=V_{p}$ for a non-dispersive medium ( in a non-dispersive medium all the waves travel with phase velocity). ➢ $V_{g}< V_{p}$ for normal dispersive medium ➢ $V_{g}> V_{p}$ for anomalous dispersive media. Dispersive medium → The medium in which the

Energy distribution spectrum of black body radiation

Description→ The energy distribution among the different wavelengths in the spectrum of black body radiation was studied by Lummer and Pringsheim in 1899. There are the following important observations of the study. The energy distribution in the radiation spectrum of the black body is not uniform. As the temperature of the body rises the intensity of radiation for each wavelength increases. At a given temperature, the intensity of radiation increases with increases in wavelength and becomes maximum at a particular wavelength with further in increases wavelength the intensity of radiation decreases. Energy distribution in the spectrum of black body radiation The points of maximum energy shift towards the shorter wavelength as the temperature increases i.e. $\lambda _{m} \times T=constant$. It is also known as Wein’s displacement law of energy distribution. For a given temperature the total energy of radiation is represented by the are

Derivation of torque on an electric dipole in an uniform and a non-uniform electric field

Torque on an electric dipole in a uniform electric field: Let us consider, An electric dipole AB, made up of two charges $+q$ and $-q$, is placed at a very small distance $2l$ in a uniform electric field $\overrightarrow{E}$. If $\theta$ is the angle between electric field intensity $\overrightarrow{E}$ and electric dipole moment $\overrightarrow{p}$ then the magnitude of electric dipole moment → $\overrightarrow{p}=q\times\overrightarrow{2l}\qquad(1)$ Force exerted on charge $+q$ by electric field $\overrightarrow{E}$ → $\overrightarrow{F_{+q}}=q\overrightarrow{E}\qquad(2)$ Here in the above equation(2), the direction of $\overrightarrow{F_{+q}}$ is along the direction of $\overrightarrow{E}$ Force exerted on charge $-q$ by electric field $\overrightarrow{E}$ → $\overrightarrow{F_{-q}}= q\overrightarrow{E}\qquad(3)$ Here in the above equation(3) the direction of $\overrightarrow{F_{-q}}$ is in the opposite direction of $\

Electric Dipole and Derivation of Electric field intensity at different points of an electric dipole

Electric Dipole: An electric dipole is a system in which two equal magnitude and opposite point charged particles are placed at a very short distance apart. Electric Dipole Moment: The product of magnitude of one point charged particle and the distance between the charges is called the 'electric dipole moment'. It is vector quantity and the direction of electric dipole moment is along the axis of the dipole pointing from negative charge to positive charge. Electric Dipole Let us consider, the two charged particle, which has equal magnitude $+q$ coulomb and $-q$ coulomb is placed at a distance of $2l$ in a dipole so the electric dipole moment is → $\overrightarrow{p}=q\times \overrightarrow{2l}$ Unit: $C-m$ Or $Ampere-metre-sec$ Dimension: $[ALT]$ Electric field intensity due to an Electric Dipole: The electric field intensity due to an electric dipole can be measured at three different points: Electric