Physics Vidyapith ✍️

History of the Special Theory Relativity (Brief Overview): Special relativity is commonly attributed to Albert Einstein’s 1905 papers. That is certainly justifiable. However, Einstein swiped the ideas of relativity from Henri Poincare, (who developed and named the principle of relativity in $1895$ and a mass-energy relation in $1900$), without giving him any credit or even mentioning his name. He may also have swiped the underlying mathematics he used from Lorentz, (who is mentioned, but not in connection with the Lorentz transformation.) However, in the case of Lorentz, it is possible to believe that Einstein was unaware of his earlier work if you are so trusting. Before you do, it must be pointed out that a review of Lorentz’s $1904$ work appeared in the second half of February $1905$ in Beibl¨atter zu den Annalen der Physik. Einstein was well aware of that journal since he wrote $21$ review journals for it himself in $1905$. Several journals were in the very next issue after t

Derivation of variation of mass with velocity: Consider two systems of reference (frame of reference) $S$ and $S’$. The frame $S’$ is moving with constant velocity $v$ relative to frame $S$. Let two bodies of masses $m_{1}$ and $m_{2}$ be traveling with velocities $u’$ and $-u’$ parallel to the x-axis in the system $S’$. Suppose the two bodies collide and after collision coalesce into one body. The principles of conservation of mass and of momentum also hold good in relativity same as in classical mechanics. So now apply the principle of conservation of momentum. $m_{1}u_{1}+m_{2}u_{2}=\left ( m_{1}+m_{2} \right )v\qquad(1)$ Apply the law of addition of velocities, the velocities $u_{1}$ and $u_{1}$ in the system $S$ corresponding to $u’$ and $-u’$ in frame $S’$ are given by $\rightarrow$ $u_{1}= \frac{u'+v}{1+\frac{u'v}{c^{2}}}\quad or \quad u_{2}= \frac{-u'+v}{1-\frac{u'v}{c^{2}}}\qquad(2)$ Now substitute the value of $u_{1}$ and $u_{1}$

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: 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: 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

Addition of Velocities: Let us consider two frames $S$ and $S'$, frame $S'$ is moving with constant velocity $v$ relative to frame $S$ along the positive direction of the X-axis. Let us express the velocity of the body in these frames. Suppose that body moves a distance $dx$ in time $dt$ in frame $S$ and through a distance $dx'$ in the time $dt'$ in the system $S'$ from point $P$ to $Q$. Then Addition of velocity $\frac{dx}{dt} = u\qquad \frac{dx'}{dt'}= u'\qquad (1)$ From Lorentz's inverse transformation $x=\alpha\left ( x'+vt' \right )\qquad(2)$ $t=\alpha '\left ( t'+\frac{v\cdot x'}{c^{2}} \right )\qquad(3)$ Differentiate the equation $(2)$ and equation $(3)$ with respect to $t'$ $\frac{dx}{dt'}= \alpha \left ( \frac{dx'}{dt'}+v \right )\qquad(4)$ $\frac{dt}{dt'}= \alpha ' \left ({1}+\frac{v}{c^{2}} \cdot \frac{

Einstein’s Mass-Energy Relation: Einstein's mass energy relation gives the relation between mass and energy. It is also knows as mass-energy equivalence principle. According to Newtonian mechanics, Newton’s second law $f=\frac{dP}{dt}$ Where $P$ is the momentum of the particle. So put $P=mv$ in above equation: $f=\frac{d}{dt}\left ( mv \right )\quad\quad (1)$ According to theory of relativity, mass of the particle varies with velocity so above equation $(1)$ can be written as: $f=m \frac{dv}{dt}+v\frac{dm}{dt}\quad\quad (2)$ When the particle is displaced through a distance $dx$ by the applied force $F$. Then the increase in kinetic energy $dk$ of the particle is given by $dk= Fdx\quad\quad (3)$ Now substituting the value of force $F$ in equation $(3)$ $dk =m\frac{dv}{dt}\cdot dx+v\frac{dm}{dt}\cdot dx \quad (4) $ $dk=mv\cdot dv +v^{2}\cdot dm \:\: (5) \: \left \{ \because \frac{dx}{dt}=v \right \}$ The variation of mass with velocity equation

Concept of Simultaneity (Relative character of Time ): The interval aren't the same for two observes in relative motion. This cause an important incontrovertible fact that two events that appear to happen simultaneously to at least one observer are not simultaneous to another observer in relative motion. Suppose two events occur (or two-time bombs explode) at different places $x_{1}$ and $x_{2}$ but at the same time $t_{0}$ with respect to an observer in a stationary frame (or on the ground). The situation of the different to an observer in moving frame $S'$ or to a pilot of a spaceship moving with velocity $v$ relative to stationary frame $S$ (or ground). To him, according to Lorentz transformation for time. The explosion at $x_{1}$ occurs at $t'_{1} = \frac{t_{0} -x_{1}\frac{v}{c^{2}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ $ x_{2}$ occurs at $t'_{2} = \frac{t_{0}- x_{2}\frac{v}{c^{2}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ Hence the two events (explosions)

It is assumed that space is continuous and the motion of particles in space can be described by their position at different instants of time. The position of a particle is known as a point in space. These points are described by the coordinate system in space. When point (position of particle in space) and the time are taken together then it is called an Event . The coordinate system of a particle which describe the position of any particle relative to it, then such coordinate system is known as Frame of Reference or System of Reference . Absolute Space: The absolute space is those frame of reference relative to which every motion and position should be measured. Types of Frame of Reference: According to the motion of particles frame of reference is divided into two categories Inertial frame of reference Non-inertial frame of reference Inertial frame of reference: