UNL - Department of Physics and Astronomy Preliminary Examination - Day Thursday, May, 28 This test covers the topics of Classical Mechanics (Topic ) and Electrodynamics (Topic 2). Each topic has 4 A questions and 4 B questions. Work two prolems from each group. Thus, you will work on a total of 8 questions today, 4 from each topic. Note: If you do more than two prolems in a group, only the first two (in the order they appear in this handout) will e graded. For instance, if you do prolems A, A3, and A4, only A and A3 will e graded. WRITE YOUR ANSWERS ON ONE SIDE OF THE PAPER ONLY
Preliminary Examination - page 2 Classical Mechanics Group A - Answer only two Group A questions A A neutron star consists of neutrons ound together y gravitational attraction and has an extremely high density ρ. What is the maximum angular frequency of rotation if a mass is not to fly off the surface? Give your answer in terms of the density of the star and some constants. You may assume a constant density and a spherical star. A2 In a homonuclear nole gas dimer ( Ne, Ar, etc.), the repulsive/attractive alance etween the two atoms is well represented y the Lennard-Jones potential, given y 2 2 2 6 Vr ( ) = 4 ε ( σ/ r) ( σ/ r). a. Find the equilirium position.. When m is the mass of each atom in the dimer, find the frequency ω of small oscillations aout the equilirium position. A3 A point particle of mass m traces out a complicated path given y x at y t z ct 3 2 =, =, and =. a. Find the angular momentum L as a function of time. Find the torque N as a function of time. A4 Consider two particles of equal mass m. One is stationary and the other moves with velocity v, striking the stationary particle. After the collision the particles oth move, with momenta p and p. Assuming a perfectly elastic collision, what is the angle etween particles velocity F 2F vectors after the collision?
Preliminary Examination - page 3 Classical Mechanics Group B - Answer only two Group B questions B A pendulum consists of a rod of zero mass and a small oject of mass m attached to it at the end. The angle the rod makes with the vertical is called θ, as shown. a. Write the Lagrangian of the system.. Write the Hamiltonian of the system. c. What is the generalized momentum p θ? d. Write the equation of motion. e. If, at time t =, we have θ = θ and θ =, what is the maximum speed at some later time? B2 B3 A ox of mass m is pushed against a spring of negligile mass and force constant k, compressing it a distance x. The ox is then released and travels up a ramp that is at an angle α aove the horizontal. The coefficient of kinetic friction etween the ox and the ramp is µ, where µ <. The ox is still moving up the ramp after traveling some fixed distance s> x k along the ramp. Calculate the angle α for which the speed of the ox after traveling this distance s is a minimum. k
Preliminary Examination - page 4 B4 Consider a car of mass m that is accelerating up a hill, as shown in the figure. The road makes an angle θ with the horizontal, as shown. An automotive engineer has measured the magnitude of the total resistive force 2 to e f = 28 +.7v, where f is in N and T T v is the speed in m/s. a. Determine the power the engine must deliver to the wheels as a function of speed and acceleration.. The mass of the car is 45 kg, and the hill has θ =. When the car accelerates with a constant acceleration a =. m/s from speed m/s to 27 m/s, what is the 2 output power as function of time?
Preliminary Examination - page 5 Electrodynamics Group A - Answer only two Group A questions A Two thin plates of infinite area and made of insulating material are on either side of the origin and a distance d away from it. They carry uniformly-distriuted surface change densities σ and a σ as shown in the figure. Find the electric potential difference etween the two plates. A2 An electric dipole consists of two charges of equal magnitude and opposite sign separated y a distance 2a, as shown in the figure. The dipole is along the x axis and is centered at the origin. a. Calculate the potential V and the electric field E on the x axis anywhere etween x the two charges (so for a< x< a). Give expressions for V and E on the x- x axis very far from the dipole ( x a). A3 The magnetic field etween the poles of the electromagnet in the figure is uniform at any time, ut its magnitude is increasing at the rate of.2 T/s. The area of the conducting loop in the field is 2 cm 2. The total resistance of the circuit, including the meter, is 5 Ω. a. Find the induced emf and the induced current in the circuit.. If the loop is replaced y one made of an insulator, what effect does this have on the induced emf and the induced current?
Preliminary Examination - page 6 A4 A current I flows through a rectangular ar (width W and length L) of conducting material along the x direction in the presence of a uniform magnetic field B pointing in the z direction, as shown z in the figure. voltmeter a. If the moving charges are positive holes with speed v (as is x source the case, for example, in certain semiconductors) instead of electrons (which the figure shows), in which direction are these holes deflected y the magnetic field? This deflection results in an accumulation of charge on the sides of the ar, which in turn produces an electric force counteracting the magnetic one. Equilirium occurs when the two cancel.. Find the resulting potential difference (the Hall voltage, V ) etween the two sides of the H ar.
Preliminary Examination - page 7 Electrodynamics Group B - Answer only two Group B questions B A very long, solid, insulating cylinder with radius R has uniform volume charge density ρ. We then drill a cylindrical hole in the original cylinder along its entire length, as shown in the figure. The hole has radius a and its axis is at a distance from the center of the original cylinder. The geometry is such that a< < R and a< R. Find the magnitude of the electric field in the hole. B2 Consider the infinite network of identical resistors shown in the figure. Each resistor has resistance r. What is the resistance etween the points A and C? B3 A rod of length L and mass m sits at rest on two frictionless conducting rails. A constant magnetic field B is applied as shown in the figure. The resistance of the rod is R. At time t = a switch S is closed, connecting a attery with potential V to the rails. The rod egins to move. a. Find the terminal speed of the rod (after S has een closed for a very long time).. Find the speed of the rod as a function of time. c. Find the current in the loop as a function of time.
Preliminary Examination - page 8 B4 Two identical, flat, circular coils of wire each have n turns and a radius of R. The coils are arranged as a set of Helmholtz coils (see figure), parallel and with a separation of R. If each coil carries a current of I, determine the magnitude of the magnetic field at a point that is on the common symmetry axis of the coils and halfway etween them.
Preliminary Examination - page 9 Physical Constants speed of light... 8 c = 2.998 m/s electrostatic constant... k = πε = 9 (4 ) 8.988 m/f 34 3 Planck s constant... h = 6.626 J s electron mass... m el = 9.9 kg 34 Planck s constant / 2π... =.55 J s electron rest energy... 5. kev 23 Boltzmann constant... k =.38 J/ K Compton wavelength.. λ = hmc / = 2.426 pm B C el 9 elementary charge... e =.62 C proton mass... m = = 2 electric permittivity... ε = 8.854 F/m ohr... a 6 magnetic permeaility... µ =.257 H/m hartree (= 2 ryderg)... E molar gas constant... Avogadro constant... N R = 8.34 J / mol K gravitational constant... p 27.673 kg 836 2 2 el m = / ke m =.5292 Å = / m a = 27.2 ev 2 2 h el G = 6.674 m / kg s 23 = 6.22 mol hc... hc = 24 ev nm A el 3 2 Equations That May Be Helpful TRIGONOMETRY sin( α + β) = sinαcos β + cosαsin β sin( α β) = sinαcos β cosαsin β cos( α + β) = cosαcos β sinαsin β cos( α β) = cosαcos β + sinαsin β sin(2 θ) = 2 sinθ cosθ 2 2 2 2 cos(2 θ) = cos θ sin θ = 2 sin θ = 2cos θ sinαsin β = cos( ) cos( ) 2 α β α + β cosαcos β = 2 cos( α β) + cos( α + β) sinαcos β = sin( α β) sin( α β) 2 + + cosαsin β = sin( α + β) sin( α β) 2 ELECTROSTATICS q encl E n ˆ da = E = V S ε a r 2 r E dl = V( r ) V( r ) 2 V() r = 4πε q( r ) r r Work done W= qe dl = qv( ) V( a) Energy stored in elec. field: W= Ed = Q C Relative permittivity: ε = + χ r e 2 2 ε τ /2 2 V
Preliminary Examination - page Bound charges ρ = P σ = P nˆ Capacitance in vacuum Parallel-plate: A C = ε d Spherical: a C = 4πε a Cylindrical: L C = 2 πε ln( / a) (for a length L) MAGNETOSTATICS Relative permeaility: µ = + χ r m Lorentz Force: F= qe+ q( v B ) Current densities: I = J d A, I = K dl µ Id Rˆ Biot-Savart Law: Br () = 2 4π ( R is vector from source point to field point r ) R Infinitely long solenoid: B-field inside is B = µ ni (n is numer of turns per unit length) Ampere s law: B dl =µ I encl Magnetic dipole moment of a current distriution is given y Force on magnetic dipole: F= ( m B) Torque on magnetic dipole: τ= m B B-field of magnetic dipole: Br µ ( ) = 3( ˆˆ ) 3 4π r Bound currents m= I da. J K = M = M nˆ
Preliminary Examination - page Maxwell s Equations in vacuum ρ. E = Gauss Law ε 2. B = no magnetic charge 3. B E = t Faraday s Law 4. E B= µ J+ εµ t Ampere s Law with Maxwell s correction Maxwell s Equations in linear, isotropic, and homogeneous (LIH) media. D = ρ Gauss Law f 2. B = no magnetic charge B 3. E = Faraday s Law t D 4. H= J + Ampere s Law with Maxwell s correction f t Induction dφ B Alternative way of writing Faraday s Law: E d = dt Mutual and self inductance: Φ = M I, and M = M ; Φ= LI 2 2 2 2 2 2 Energy stored in magnetic field: W = µ B dτ LI d 2 = = 2 2 A I V
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Preliminary Examination - page 3 CARTESIAN AND SPHERICAL UNIT VECTORS xˆ = (sinθcos φ) rˆ+ (cosθcos φ) θˆ sin φˆ yˆ = (sinθsin φ) rˆ + (cosθsin φ) θˆ + cos φˆ zˆ = cosθ rˆ sin θˆ INTEGRALS + x x e 2 n x dx = π /2 n! dx = n+ /2 ( ) ( x + ) dx = ln x+ x + 2 2 /2 2 2 2 2 ( x + ) dx = arctan( x/ ) 2 2 3/2 x ( x + ) dx = 2 2 2 x + x + arctan( x/ ) 2 2 2 2 2 ( x + ) dx = x + 3 2 xdx 2 2 = 2 2 ln ( x + ) x + 2 2 dx x = 2 2 2 ln 2 2 x( x + ) 2 x + dx ax = 2 2 2 ln ax 2a ax + ax = artanh a