Construction and Working of Nuclear Reactor or Atomic Pile

A nuclear reactor is a device within which a self-sustaining controlled chain reaction is produced by fissionable material. it is thus a source of control energy that is utilized for several useful purposes. The reactor has some important part which is given below:

  1. Fuel: The fassionable material such as Uranium-235 and Plutonium-239 known as fuel. These materials play an important role in operating the nuclear reactor.

  2. Moderator: It slows down the neutrons to thermal energies through the elastic collision between its nuclei and fission neutrons. Thermal neutrons have a very high probability of fissioning Uranium-235 nuclei. Examples: heavy water graphite beryllium oxide. Heavy water is the best moderator.

  3. Control Rods: These rods are used to control the fission rate in the reactor. these Rods are fixed in the reactor walls. These rods are made up of the material of cadmium and Boron. These materials are good absorbers of slow neutrons. Therefore when the rods are pushed into the reactor, the fission rate decreases, and when they are pulled out, the fission grows.

  4. Coolant: The coolant is used to remove the heat from the reactor which is produced inside the reactor. For this purpose air-water or CO2 is passed through the reactor.

  5. Shield: various type of intense rays are emitted from the reactor, which may be injurious to the people working near the reactor. To protect them from this radiation, thick concrete walls are erected around the reactor.

  6. Safety device: In case of an accident or other emergency, a special set of control rods, called " shut-off rods" enter the reactor automatically. They immediately absorb the neutrons so that the chain reaction stops entirely.

Construction:

It is made up of a large number of uranium rods which are placed in calculated geometrical lattice between layers of pure graphite ( moderator) blocks. To prevent oxidation of Uranium, the rods are covered by close-fitting aluminum cylinders. The control rods are so inserted within the lattice that they will be raised or lowered between the Uranium rods whenever necessary. The whole reactor is encircled by a concrete shield.

Working:

The actual operation of the reactor is started by pulling out the control rods so that they do not absorb many neutrons. Then, the stray neutrons, which are always present in the tractor, start fissioning the U-235 nuclei. In each fission, two or three fast neutrons are produced. These neutrons repeatedly strike the moderator and slowed down. Then, These neutrons start fissioning the U-235 nuclei. So, a chain-reaction of fission starts. The number of neutrons, which is produced in fissioning, is controlled by pushing the cadmium rods into the reactor. These rods absorb a number of the neutrons. Thus, the energy produced in the reactor is kept under control to avoid any explosion. The coolant is pond pumped through the reactor to carry away The Heat generated by the fission of uranium nuclei. the hot CO2 passes through the heat exchanger and convert cold water into steam. This steam is used to operate turbines for generating electricity.

Einstein’s Mass Energy Relation Derivation

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

$m=\frac{m_{\circ}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\quad\quad (6) $

Square both sides in above equation:

$m^{2}=\frac{m_\circ^{2}}{1-\frac{v^{2}}{c^{2}}}$

$m^{2}c^{2}-m^{2}v^{2}=m_\circ ^{2}c^{2}$

Differentiate the above equation which can be written as

$2mdm\cdot c^{2}- 2m \cdot dm \cdot v^{2}-2v\cdot dv\cdot m^{2}=0$

$c^{2}dm-v^{2}dm-vm\cdot dv$

$c^{2}dm=v^{2}dm+mv\cdot dv \quad\quad (7)$

Now substitute the value of $dk$ from equation $(5)$ in equation $(7)$. So above equation can be written as:

$dk = c^{2}dm$

Now consider that the particle is at rest initially and by the application of force it acquires a velocity $v$. The mass of body increase from ${m_{\circ}}$ to $m$. The total kinetic energy acquired by the particle is given by

$dk = \int_{m_\circ}^{m}c^{2}\cdot dm$

$k = c^{2}\left ( m-m_{\circ} \right )$

$k = mc^{2} - m_\circ c^{2}$

$k+m_\circ c^{2} = mc^{2}$

Where $k$ is the kinetic energy of the particle and $m_\circ c^{2}$ is the rest mass-energy of the particle. So The sum of these energies is equal to the total energy of the particle $E$. So

$E= m c^{2}$

Where $E$ is the total energy of the particle.

The above equation is called the mass energy equivalence equation.

Concept of Simultaneity in Special Relativity

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) that occur simultaneously to one observer in the stationary frame are separated to another observer in a moving frame by an interval of

$t'_{2}-t'_{1}=\frac{\left ( x_{1}-x_{2} \right )\frac{v}{c^{2}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$

Therefore, the principle of simultaneity in relativity is an absolute concept for two events. It depends on an observer or a frame of reference. The effect is not due to the time dilation. Hence, we conclude that there is no such thing as “absolute time” which is the same for all observers.

“Time is relative and it varies for all observers in relative motion.”

Mean Value and Root Mean Square Value of Alternating Current

Derivation of the Mean Or Average Value of Alternating Current:

Let us consider alternating current $i$ propagating in a circuit, then the average value of the current.

$ i_{mean}=\frac{1}{\left ( \frac{T}{2} \right )}\int_{0}^{\frac{T}{2}}i \:dt \qquad (1)$

Where $\quad i = i_{0}sin \omega t \quad(2)$

Now substitute the value of current $i$ in above equation $(1)$

$ i_{mean}= \frac{2}{T}\int_{0}^{\frac{T}{2}}i_0. sin \omega t \cdot dt$

$ i_{mean}= \frac{2.i_{0}}{T}\int_{0}^{\frac{T}{2}}\sin \omega t \cdot dt$

$ i_{mean}= \frac{2 i_{0}}{T}[\frac{-cos\:\omega t}{\omega} ]_{0}^{\frac{T}{2}}$

The value of $\omega$ is $\frac{2 \pi}{T}$ i.e $\omega=\frac{2\pi}{T}$

$ i_{mean}= \frac{2 i_{0}}{T \left (\frac{2\pi}{T} \right )}\left [ -cos \left (\frac{2 \pi}{T} \right ) \left ( \frac{T}{2} \right ) \\ \qquad \qquad \qquad +cos0^\circ \right ] $

$ i_{mean}= \frac{i_{0}}{\pi}\left [ - cos\pi+cos0^{\circ} \right ]$

$ i_{mean}=\frac{i_{0}}{\pi}\left [1+1 \right ]$

$ i_{mean} =\frac{2 i_{0}}{\pi}$

$ i_{mean} = 0.637\: i_{0}$

From the above equation, we can conclude that the mean or average value of alternating current for one cycle is $0.637$ times or $63.7 \%$ of its peak value $i_{0}$.

Root mean square value of Alternating Current:

Let us consider a current $i$ propagating in a circuit, then the mean square value of alternating current.

$ \left (i_{mean} \right )^{2} = \frac{1}{T}\int_{0}^{T}i^{2} \cdot dt \quad(1)$

$ where\quad i= i_{0} \dot sin\omega t\quad (2)$

Now substitute the value of current $i$ in above equation $(1)$

$ \left ( i_{mean} \right )^{2} = \frac{1}{T}\int_{0}^{T} \left ( i_{0} \:sin\omega t\right )^{2} dt $

$ \left (i_{mean} \right )^{2} = \frac{i_{0}^2}{T}\quad \int_{0}^{T} sin^{2}\omega t \cdot dt$

$ \left (i_{mean} \right )^{2} = \frac{i_0^{2}}{T}\int_{0}^{T} \frac{1-cos2\omega t}{2}\ dt$

$ \left (i_{mean} \right )^{2} = \frac{i_o^{2}}{2T}\int_{0}^{T}\left ( 1-cos2\omega t \right )dt$

$ \left (i_{mean} \right )^{2} = \frac{i_{0}^{2}}{2T}\left [ \left ( t \right )_{0}^{T} - \left ( \frac{sin2 \omega t}{2 \omega} \right )_{0}^{T} \right ] $

$ \left (i_{mean} \right )^{2} = \frac{i_0^{2}}{2T}\left [ \left ( T-0 \right ) \\ \qquad\qquad\qquad -\frac{1}{2\omega}\left ( sin2\omega T-sin 0^{\circ} \right ) \right ]$

$ \left (i_{mean} \right )^{2} = \frac{i_0^{2}}{2T}\left [ \left ( T-0 \right ) \\ \qquad\qquad\qquad -\frac{1}{2\omega}\left ( sin2\omega T-sin0^{\circ} \right ) \right ]$

$ \left (i_{mean} \right )^{2} = \frac{i_0^{2}}{2T}\left [ T-\frac{1}{2\omega}\left ( sin4\pi \\ \qquad \qquad\qquad -sin0^{\circ} \right ) \right ]$

$ \left (i_{mean} \right )^{2} = \frac{i_0^{2}}{2T}\left [ T-\frac{1}{2\omega}\left ( 0-0 \right ) \right ]$

$ \left ( i_{mean} \right )^{2}=\frac{i_0^{2}}{2}$

$ \left (i_{mean} \right )^{2} = \frac{i_0^{2}}{2}$

So root mean square value of the above equation:

$ i_{rms} = \sqrt{i_{mean}^{2}}$

$i_{rms} = \frac{i_{0}}{\sqrt{2}}$

$i_{rms} = 0.707\:i_{0}$

Thus, the root mean square value of an alternating current is $0.707$ times or $70.7 \%$ of the peak value.

Assumptions of Planck’s Radiation Law

Planck in 1900 suggested the correct explanation of the black body radiation curve. They gave the following assumption →

  1. A chamber contains black body energy radiation and simple harmonic oscillators (atoms of Wall, i.e. Black lamp & Platinum coating inside wall, behave as oscillators or resonators) of molecular dimensions which can vibrate with all possible frequencies.

  2. The frequency of energy radiation emitted by an oscillator is the same as the frequency of its vibration.

  3. An oscillator cannot emit or absorb the energy in a continuous manner it can emit or absorb energy in a small unit (packet) called Quanta.

If an oscillator is vibrating with a frequency $ \nu $ it can only radiate in quanta of magnitude $h\nu $ i.e. “The oscillator can have only discrete energy value $E_{n}$ ” given by–

$E_{n}=nh\nu$

Where
$n$ – an integer
$h$– Planck ’s constant and the value is $6.626\times10^{-34} J-s$

The average energy of Planck’s oscillator of frequency $\nu$ -

$E_{\lambda}d\lambda = \frac{8\pi hc}{\lambda ^{5}} \frac{d\lambda }{(e^{\frac{hc}{\lambda kT}}-1)}$

$E_{\nu}d\nu= \frac{8\pi h\nu^{3}}{c^{3}}\frac{d\nu }{(e^{\frac{h\nu}{kt}}-1)}$

This assumption is most revolutionary in character. This implies that the exchange of energy between radiation and matter (Black lamp or platinum Coating ) cannot take place continuously but are limited to a discrete set of value $ 0, h\nu, 2h\nu, 3h\nu,------ nh \nu $.

Heisenberg uncertainty principle

If the x-coordinate of the position of a particle is known to an accuracy of $\delta x$, then the x-component of momentum cannot be determined to an accuracy better than $\Delta P_{x}\approx \frac{\hbar }{\Delta x}$.

$\Delta P_{x}. \Delta x\approx \hbar$

The above inequality must be satisfied

$\Delta P_{x}. \Delta x\geqslant \hbar$

Where $\hbar $ - Planck’s Constant

This is the Uncertainty principle with macroscopic objects. Exact statement of the Uncertainty principle →

The product of the uncertainties in determining the position and momentum of the particle can never be smaller than the number of the order $\frac{\hbar }{2}$.

$\Delta P_{x}. \Delta x\geqslant \frac{\hbar}{2}$

Where $\delta x$ and $\delta P $ are defined as the root mean square deviation from their mean values.

The Uncertainty principle can also describe by the following formula →

$\Delta x.\Delta p_{x}\approx \frac{\hbar}{2}$

$\Delta x.\Delta p_{x}\geqslant \frac{\hbar}{2}$

$\Delta x.\Delta p_{x}\geqslant \frac{h}{4\pi }$

Expression for $y$ and $z$ component →

$\Delta y.\Delta p_{y}\geqslant \frac{h}{4\pi }$

$\Delta z.\Delta p_{z}\geqslant \frac{h}{4\pi }$

The uncertainty relation between energy and time →

$\Delta E.\Delta t\geqslant \frac{h}{4\pi }$

$\Delta E.\Delta t\geqslant \frac{\hbar }{2 }$

The uncertainty relation between momentum and Angular Position→

$\Delta L.\Delta \theta \geqslant \frac{h }{4\pi }$

$\Delta L.\Delta \theta \geqslant \frac{\hbar}{2}$

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