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Romanian Master of Physics 2016

Q1. Compressible fluids

The study of gases flow uncovers many interesting phenomena which have a myriad of applications, starting from boilers to airplanes and rockets. To simplify the calculations, in this problem the following assumptions will be adopted: - The gas is ideal; - The gas flow is steady and non-turbulent; - The processes taking place in the flowing gas are adiabatic; - The gas flow speed is much less than the speed of light; - The gas flow is uniform and one-dimensional (axisymmetric); - The effect of gravity is negligible. The constants useful in this problem are: - the molar mass of air, ; the ideal gas constant, . A. Bernoulli’s equation Bernoulli’s equation is the mathematical form of the law of conservation and transformation of energy for a flowing ideal gas. It bears the name of the Swiss physicist Daniel Bernoulli (1700 - 1782), who derived it in 1738. The easiest way to obtain this equation is to follow a fluid particle (a volume element of the fluid) in its way on a streamline. A Perform the energy balance between two points in the flowing gas, knowing the parameters and , and derive the equation that connects 1.5 p these variables. The adiabatic exponent of the gas is also known. The parameter is the gas pressure, its density, and its speed.

B. Propagation of a perturbation in a flowing gas If the pressure in a layer of a macroscopically motionless gas system suddenly increases (by heating or rapid compression), the layer will begin to expand, compressing the adjacent layers. This pressure disturbance will be thus transmitted by contiguity as an elastic wave through the gas. B1. Speed of the perturbation The speed c of this wave is the speed of its wavefront (the most advanced surface, the points of which oscillate in phase and the thermodynamic parameters of which have the same value). If in the reference frame of the unperturbed gas the process of the wave propagation is nonsteady (the gas parameters in any point vary with time), in the reference frame of the wavefront the process will be steady, so the simple equations for a steady state can be applied. B1 Derive the mathematical expression for the speed of the wavefront, taking into account that the thermodynamic parameters of the unperturbed gas are 1.5 p while those “behind” the wavefront are

Theoretical Problem No. 3 Page 1 from 14

Romanian Master of Physics 2016

B.2 Sound waves Sound waves are waves of weak disturbances ( and ) that travel fast enough, their speed being of the order of hundreds of meters per second. Due to this, the gas compressions and rarefactions can be considered as adiabatic, the adiabatic exponent being . B2 Using the result from B1, obtain the mathematical expression for the sound speed in the gas and, using Bernoulli’s eq., derive the relation between the flow speed 0.5 p at a given point in the gas and the local sound speed.

B.3 Mach’s number For classifying the speed performances of bodies in a fluid (e.g. aircrafts), as well as the flow regimes of fluids, the Swiss aeronautical engineer Jakob Ackeret (1898 – 1981) – one of the leading authorities in the 20th century aeronautics – proposed in 1929 that the ratio of the body or of the fluid’s speed v and the local sound speed c in that fluid to be called Mach’s number after the name of the great Czech (then in the Austrian empire) physicist and philosopher Ernst Mach (1838 – 1916). Primarily, the value of this non-dimensional quantity delimitates the incompressible from the compressible behavior of a flowing fluid, in aeronautics this limit being settled to . B3.1 Find the relative variation of the gas density as a function of Mach’s number, when its motion is slowed down to a stop, its initial velocity being v ? c , and 0.5 p calculate its maximum value for a flow to be considered incompressible. B3.2 The pressure at the nose of an aircraft in flight was found to be and the speed of air relative to the aircraft was zero at this point. The pressure 0.5 p and temperature of the undisturbed air were and respectively. The adiabatic exponent for this temperature is . Find the speed and the Mach number of the aircraft.

B3.3 When a gas is flowing through a pipe, it exerts a friction force on the fluid, which is not always negligible. If at the entrance of such a pipe the static pressure in the flowing fluid is and the Mach number is , while at the exit , find the expression and the numerical 1.0 p value of the force with which the fluid is acting on the pipe. The adiabatic exponent is , the constant cross section of the pipe is , and the relative increase of the gas temperature through the pipe is .

Theoretical Problem No. 3 Page 2 from 14

Romanian Master of Physics 2016

C. Shock waves There are two types of acoustic waves in a gas: sound waves and shock waves. The latter appear when a body moves in a gas with a supersonic speed (i.e. the relative speed of the body with respect to that of the gas is greater than the sound speed). At supersonic speeds, in front of the body appears a very thin layer of gas with a higher pressure, called compression shock. This kind of special acoustic waves were studied by Mach, so the envelope of such a wave is known as Mach’s cone, having the body in its apex. Passing through the compression shock, the thermodynamic parameters of the gas change abruptly. The Mach’s cone is an example of an oblique shock, but we are interested here mainly in normal shocks, for which the shock wavefront is perpendicular on the body or fluid velocity. For shock waves the pressure/density differences between the two sides of the wavefront can reach very high values. Passing through the wavefront, the thermodynamic parameters vary abruptly, with a sudden jump. This is another reason for which a shock wavefront is called a pressure or a compression shock. C.1 The shock adiabat The gas compressed by the shock wave undergoes an irreversible adiabatic process which cannot be described by Poisson’s equation. However, an equation for the shock adiabat was deduced towards the end of the XIXth century by the Scottish physicist William Rankine (1820 – 1872) and, independently by him, by the French engineer Pierre Henri Hugoniot (1851 – 1887), using the mass and energy conservation, as well as the momentum equation. The Rankine – Hugoniot equation, or the shock adiabat, relates the pressure and the density of the gas compressed by a shock wave. C1 Denoting with and the gas pressure and density in front of the compression shock (which are known), and with and the same parameters behind the shock (which are unknown), show that the pressure ratio is related with the density ratio by a relation of the form 1.5 p Find the explicit form of the coefficients , , and . The adiabatic exponent of the gas is known. Note: For simplicity, use a stream tube with a constant cross section, crossing perpendicularly the wavefront of the normal shock.

Theoretical Problem No. 3 Page 3 from 14

Romanian Master of Physics 2016

C.2 A shockwave created by an explosion An explosion creates a spherically shockwave propagating radially into still air at . A recording instrument registers a maximum pressure of wavefront passes by. The adiabatic coefficient of air for this compression shock is mass of air is C2.1 and the ideal gas constant is .

and as the shock , the molar

Determine the air temperature increase shock.

under the action of the compression

0.5 p

C2.2 Determine the Mach’s number corresponding to the speed of the shockwave. C2.3 Determine the wind’s speed fixed observer. following the shock wavefront, with respect to a

0.5 p

0.5 p

During the compression shock the gas temperature and pressure sharply increase, much more than in a quasistatic adiabatic compression. After the shock, the gas expands adiabatically, but because the slope of the adiabatic process is smaller than that of the adiabatic shock, when the gas density reaches again the initial value, its pressure is still higher than that of the unperturbed gas, . C2.4 Derive the ratios numerical values of and and at the end of the expansion process and calculate the . 0.5 p

C2.5 From this point the gas is cooling until it reaches the initial state. Assuming that for the entire cyclic process the adiabatic exponent has the same value, derive 1.0 p an expression for the entropy variation of the mass unit of air during the compression shock and calculate its numerical value.

? Assoc. Prof. Sebastian POPESCU, PhD Faculty of Physics, Alexandru Ioan Cuza University of Ia?i, ROMANIA

Theoretical Problem No. 3 Page 4 from 14

Romanian Master of Physics 2016

Q1. Compressible fluids

Answer sheet

The equation that relates

and

is

A

B1

B2

The relation between the flow speed at a given point in the gas and the local sound speed is:

Theoretical Problem No. 3 Page 5 from 14

Romanian Master of Physics 2016

B3.1

B3.2

The expression of the force is: B3.3

The numerical value of the force is:

C1

C2.1

Theoretical Problem No. 3 Page 6 from 14

Romanian Master of Physics 2016

C2.2

C2.3

Analytical expressions

Numerical values

C2.4

Analytical expression

Numerical value

C2.5

Theoretical Problem No. 3 Page 7 from 14

Romanian Master of Physics 2016

Q2. Sources, atoms, and spectra

A. Light source In its proper reference frame, a point source emits light in the form of a divergent conical beam, with the angular width of 90° (from -45° to +45° with respect to the cone axis). In a reference frame which moves towards the source with an unknown speed , the angular width of the beam is of only 60° (from -30° to +30° with respect to the same cone axis). The light speed in vacuum is . A Determine the speed of the source. 2.50 p.

B. Balmer emission spectrum The spectral resolving power of a spectrometer is . The spectrometer is used to observe the Balmer series in the emission spectrum of the hydrogen atom (the visible domain). Note: The possible mechanisms of broadening of the spectral lines (Lorentzian, Gaussian, etc.) will not be considered. B.1 Express the mathematical definition of the spectral resolving power of the instrument. Determine the highest value for the principal quantum number of the energy level for which the spectral line emitted by an atom for the transition to the level can still be distinctly resolved by the instrument, with respect with its neighbours. 0.25 p.

B.2

2.25 p.

C. Absorption spectra The energy levels of an atom are given by , where is an integer and is a positive constant. Among the adjacent spectral lines which, at room temperature, the atom can absorb, two have the wavelengths and , respectively. The elementary electric charge is , the speed of light in vacuum is c , and Planck’s constant is . C.1 Find the values of the quantum numbers the transitions. Determine the value of the constant of the energy levels implied in 3.00 p. 1.50 p. 0.50 p.

C.2 C.3

in joule and in electron-volt.

Identify the nature of the atom and justify the choice made.

? prof. Florea ULIU, PhD, University of Craiova

Theoretical Problem No. 3 Page 8 from 14

Romanian Master of Physics 2016

Q3. Sources, atoms, and spectra

Answer sheet A Final expression for the speed of the source Numerical value for the speed of the source 2.50 p.

B.1 0.25 p.

B.2 2.25 p.

C.1 3.00 p.

in joule C.2 in electron-volt

1.00 p.

0.50 p

C.3

0.50 p.

Theoretical Problem No. 3 Page 9 from 14

Romanian Master of Physics 2016

Theoretical Problem No. 3 (10 points) ”Squeezing” electrical charge carriers using magnetic fields

Plasma physics has to solve the problem of achieving devices capable of producing energy on a large scale through fusion. There are no issues related to the feasibility of the scientific method - the process can be observed as it happens in stars. But there are many technological feasibility problems primarily related to heating the plasma, and controlling it, at temperatures like the ones in the Sun. Physical conditions of the nuclear fusion cannot be achieved in containers. Maintaining plasma localized in limited volume can be accomplished using magnetic fields. The first two tasks of the problem require analyzing several situations in which the movement of charged particles is limited by external magnetic fields. The third task asks you to study the confinement of electrical charge carriers through their own magnetic field. When solving the problem you may rely on the following numerical values: elementary electric charge e ? 1.6 ? 10 ?19 C , mass of the electron m ? 9.1? 10 ?31kg , the magnetic permeability of vacuum ?0 ? 4? ? 10?7 F ? m?1 .

Task No. 1

A very long metallic cylinder (a perfect electric conductor), having the length L and the radius R , ?L ?? R ? rotates at constant angular speed ? around its axis of symmetry. ? The cylinder is located in a homogeneous magnetic field whose induction B is parallel to the axis of symmetry of the cylinder. The mass of electron is m and his electric charge is ? e . Let the dielectric permittivity of the material from which the cylinder is made be ? . 1.i. For a stationary situation determine the expression of the bulk density of electric charge ? into the cylinder at a distance r from its axis of symmetry ?0 ? r ? R ? . Express the result as function of B, e, m ,? and ? . (1.00p) 1.ii. Determine the expression of angular velocity ? 0 so that the bulk charge density is zero at any point of the cylinder. Express the result as function of B, e and m . (0.20p) 1.iii. Evaluate the possibility of the practical realization of a zero bulk charge density in any point of the metallic cylinder, when an experiment of the type described in question is carried out in the Earth's ,82 ? 10?5T . Briefly argue the answer. magnetic field in a place where the magnetic field induction is B ? 1 (0.30p)

Task No. 2

In a vacuum chamber there is a long, straight wire of negligible thickness made from a material with high electrical conductivity. An electrical current with the intensity of I ? 10A passes through the wire. Electrons are sent on a direction perpendicular on the wire, towards the wire. Their motion starts at the distance r0 from the wire, with an initial speed v 0 much smaller than speed of light. The electrons cannot approach the wire at a distance less than r0 2 . Consider two reference frames – one being the laboratory system S.L. and the other being a mobile system S.M. that runs parallel to the wire with a constant speed v 0 in the direction in which the current flows through the wire. Neglect the magnetic field of Earth. 2.i. Deduce the expressions of induction of the ? magnetic ? field produced by the current flowing through the wire in each of the two reference frames BS.L. (r ) , BS.M . (r ) . (1.00p)

Theoretical Problem No. 3 Page 10 from 14

Romanian Master of Physics 2016

2.ii. the expression of the difference between Lorentz forces in the two reference systems, ? Determine ? FS.L. ?r ? ? FS.M. ?r ? , forces acting on the electron sent to the wire. Express the result in terms of e, ?0 , I, r and v 0 . (1.50p) ? 2.iii. State the values of the electron velocity’s v (r0 2) components in both reference frames. (1.00p) 2.iv. Determine the expression of the speed v 0 as function of e, m, I, ?0 . Calculate the numerical value of v 0 . (1.50p) 2.v. Deduce the expression of the maximum distance from the wire D at which one can find the electron, as function of r0 , when the electron moves away from wire on a direction perpendicular on it. (0.50p)

Task No. 3

A cylindrical column of plasma having the radius R and the length L is generated in a vacuum chamber. The plasma appears as result of ionization of a gas, such that the concentrations of electrons ne (r ) and ions ni (r ) are equal at every point ni (r ) ? ne (r ) ? n(r ) ; the common value n(r ) depends only on the distance between the point and the axis of symmetry of the plasma cylinder. It is assumed that the plasma is in a stationary state, and therefore all its macroscopic characteristics are independent on time. It may be admitted that the temperature T of plasma is the same at every point of the column and that at these temperature the parameters describing plasma abide the perfect gas law. The electric charge of electron is ? e and the ions are monovalent, carrying an electric charge e . Consider as known the magnetic permeability of plasma ? and Boltzmann’s constant k B . Between the electrodes at the ends of the plasma column passes through the plasma an electrical current characterized by a constant density j (r ) ? j . Consider an elementary portion of the hollow cylinder having the radiuses r and r ? dr as in joined figure. Elementary portion has a height equal to the unit. In the annulus (circular crown) representing the cross section of the plasma column the elementary portion covers the angle d? .

? 3.i. Write the expressions of the forces acting on this elementary volume of plasma ( Fp due to pressure ? p (r ) in the column of plasma, Fe due to the interaction with electrical charges carriers that there are in ? plasma , Fm due to the interaction with magnetic field of electrical current flowing through plasma). Write the equation describing the equilibrium of the considered elementary portion of plasma. (1.20p)

3.ii. Deduce the expression of pressure p (r ) in a point of the column of plasma. Express the answer as function of r , j , R and magnetic permeability ? . (1.00p) 3.iii. Determine the expression of the number of particles N carrying electrical charge in the column of plasma as function of L , I , T , ? ?i k B . (0.80p)

? Topic proposed by: Dr. Delia DAVIDESCU Dr. Adrian DAFINEI

Theoretical Problem No. 3 Page 11 from 14

Romanian Master of Physics 2016

ANSWER SHEET

Theoretical Problem No. 3 (10 points) ”Squeezing” electrical charge carriers using magnetic fields Task No. 1

1.i. The expression of the bulk density of electric charge ? into the cylinder at a distance r from its axis of symmetry

1.00p

1.ii. The expression of angular velocity ? 0 so that the bulk charge density is zero at any point of the cylinder 0.20p

1.iii. Evaluate the possibility of the practical realization of a zero bulk charge density in any point of the metallic cylinder. Briefly argue the answer

0.30p

Task No. 2

2.i. The expressions of induction of the magnetic field produced by the current flowing through the ? wire in each ? of the two reference frames BS.L. (r ) and BS.M . (r )

1.00p

2.ii. The expression of the difference between Lorentz forces ? in the ? two reference systems, FS.L. ?r ? ? FS.M. ?r ?

1.50p

Page 12 of 14

Romanian Master of Physics 2016

? 2.iii. State the values of the electron velocity’s v (r0 2) components in both reference frames

1.00p

2.iv. The expression of the speed v 0 as function of e, m, I, ?0 and the numerical value of v 0 2.v. The expression of the maximum distance from the wire D at which one can find the electron, as function of r0 , when the electron moves away from wire on a direction perpendicular on it

1.50p

0.50p

Task No. 3

3.i. The expressions of the forces acting on this elementary volume of plasma

0.80p

The equation describing the equilibrium of the considered elementary portion of plasma

0.40p

3.ii. The expression of pressure p (r ) in a point of the column of plasma

1.00p

Page 13 of 14

Romanian Master of Physics 2016

3.iii. The expression of the number of particles N carrying electrical charge in the column of plasma

0.80p

Page 14 of 14

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