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• Derive Taylor expansion of exponential and for $x=1$ graphically show sequence of partial sums of series \sum_{$k=1}^{∞}1⁄k!$ with series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$.

Using the same method compare series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$ and series of partial sums of series \sum_{$k=1}^{∞}x^{k}⁄k!$, therefore series {\sum_{$k=1}^{n}x^{k}⁄k!}_{n=1,2,\ldots}$, now for $x=-1$.

• The second task is to find concentration of electrons and positrons on temperature with total charge $Q=0$ in empty and closed cavity (you can choose value of $Q.)$ Further calculate dependence of ration of internal energy $U_{e}$ of electrons and positrons to the total internal energy of the system $U$ (e.g. the sum of energy of electromagnetic radiation and particles) on temperature and find value of temperature related to some prominent temperature and ratios (e.g. 3 ⁄ 4, 1 ⁄ 2, 1 ⁄ 4, …; can this ratio be of all values?).

Put your results into a graph – you can try also in 3-dimensions.

To get the calculation simplified, it could help to take some unit-less entity (e.g. $βE_{0}$ instead of $β$ etc.).

• Solve the system of differential equations for $M(r)$ and $ρ(r)$ in model of white dwarf for several well chosen values of $ρ(0)$ and for every value find the value which it get close $M(r)$ at

$r→∞$. This is probably equal to the mass of the whole star. Try to find the dependence of total weight on $ρ(0)$ and find its upper limit. Compare the result with the upper limit of mass for white dwarf (you will find it in literature or internet). Assume, that the star consists from helium.

Calculate time dependence of wave function of particle, which is located in potential $V(x)=\frac{1}{2}kx$ and which is at time $τ=0$ described by wave function

$ψ_{R}(X,0)=\exp(-((X-X_{0}))⁄4)$,

ψ$_{I}(X,0)=0$.

It is wave packet with the center not in the origin. We can tell you, that this is so called coherent stat of harmonic oscillator and wave packet should oscillate around origin with angular frequency √( $k⁄m)$ same as classical particle.

If you can calculate the previous, then you can try what will be the behaviour of wave packet of different width (e.g. denominator in exponential is different from 4) of how the behaviour will look like with different potential.

Uranium storage

Heating wire

Capacity of a cube

Estimate thickness of water needed to shield radiation from nuclear reactor of power 980 MW in proposed new block of nuclear power station Temelin. The total energy released by fission from Uranium is split following: 82% goes to kinetic energy of fragments, 6% goes to neutrinos, 6% goes to neutrons and remaining 6% goes to energy of gamma photons.

Hint: Probability of particle passing through a material into the depth $d$ is probably equal to e^{$-σnd}$, where $n=N/V$ is density of molecules of material (in our case number of water molecules in 1 m3 ) and $σ$ is effective cross section for absorption of particle on molecule of water. Unit of cross section is surface, therefore m^{2} but often is used unit barn = 100 fm2) and depends on energy of particles. The values of cross section can be found on Internets or in tables.

The typical composition of natural uranium is 0.72 % of isotope ^{235}U of halftime of 704 million years and the rest is isotope ^{238}U with half life of 4 468 million years.

While mining uranium in 1970's in Okla in equatorial Gabon, the uranium ore with relative content of isotope ^{235}U 0.44 % . This discrepancy can be explained by assuming existence of 'natural nuclear reactor' at the uranium deposits million years ago.

Calculate, how long the nuclear reaction was going on if the decay of ^{235}U was started by slow neutrons. The collision of slow neutron with the nucleus happens every 352 thousand years.

In Jet Propulsion Laboratory in California, U.S.A. in NASA laboratory the new rocket engine is under development. It uses momentum of $α-particles$ created during radioactive decay of fermium $^{257}_{100}Fm_{157}$, which mass is $m_{Fm}$ and half-life $T$. The second product is californium $^{253}_{98}Cf_{155}$. The mass of $α-particle$ is $m_{α}$, the mass of californium is $m_{Cf}$, and during the decay the energy $E$ is released. Assume, that each $α-particle$ leaves rocket in the same direction.

The space probe with above engine is in rest at the beginning and its mass is $M$, the mass of 'fuel' is also $M$. Calculate the speed of the probe $v$ after half of the fermium decays. Resulting speed calculate also for the following numerical values $E=1,106\cdot 10^{-12}J$, $M=4\;\mathrm{kg}$ a $T=100,5days$, for other values consult your table-book.

When measuring decay of neutron to electron and proton the energy of the electron was detected. How can be detected, that another particle was not created? Assume the neutron to be at rest at the beginning.

The frequency of photon emitted by the nucleus of radioactive iron is not always the same, but is slightly different (it is true also for other elements). To make thinks simpler, assume that the energy of photon in the frame connected with the resting nucleus of iron is randomly in interval ( $E_{0}-ΔE,E_{0}+ΔE)$, where $E_{0}=14,4\;\mathrm{keV}$ (keV = kiloelektronVolt), $ΔE≈10^{-8}\;\mathrm{eV}$ (1 eV = 1,602 \cdot 10^{-19} J ).

• When the photon is emitted from a stationary nucleus the nucleus acquires opposite momentum to the photon. Calculate kinetic energy of the atom and compare it with the $ΔE$.
• So called Mössbauer effect is the transfer of momentum of recoil to the crystal (whose the atom is part of). Calculate kinetic energy of the crystal (the shift in photon energy) assuming that the crystal consist of 10^{23} atoms.

Same as the emission of photon also excitation occurs. The photon can be absorbed only if its energy in the rest frame of atom is in interval ( $E_{0}-ΔE,E_{0}+ΔE)$.

• Decide if the resting atom of iron can absorb photon emitted by another resting atom of iron.
• Calculate the relative speed of two pieces of iron needed for Doppler effect to forbid the absorption of the photon in the second piece of iron. The Doppler effect is the change in the frequency $f$, of the photon when the source is coming closer to the observer at the speed $v$. The frequency is changed to

$f′=(1+v⁄c)f$.

Assume that the Mössbauer effect takes place at the emission.

Find all the needed constants in tables.