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physics
modern physics

Questions and Answers of Modern Physics

An object of mass 0.12 kg is moving at 1.80 × 108 m/s. What is its kinetic energy in joules?
When an electron travels at 0.60c, what is its total energy in mega-electron-volts?
An observer in the laboratory finds that an electron's total energy is 5.0mc2. What is the magnitude of the electron's momentum (as a multiple of mc), as observed in the laboratory?
An astronaut wears a new Rolex watch on a journey at a speed of 2.0 × 108 m/s with respect to Earth. According to mission control in Houston, the trip lasts 12.0 h. How long is the trip as measured
The rest energy of an electron is 0.511 MeV. What momentum (in MeV/c) must an electron have in order that its total energy be 3.00 times its rest energy?
An electron has a total energy of 6.5 MeV. What is its momentum (in MeV/c)?
For a non-relativistic particle of mass m, show that K = p2 / (2m).
Find the conversion between the momentum unit MeV/c and the SI unit of momentum.
Find the conversion between the mass unit MeV/c2 and the SI unit of mass.
In a beam of electrons used in a diffraction experiment, each electron is accelerated to a kinetic energy of 150 keV. (a) Are the electrons relativistic? Explain. (b) How fast are the electrons
Starting with the energy-momentum relation E2 = E02 + (pc)2 and the definition of total energy, show that (pc)2 = K2 + 2KE0 [Eq. (26-11)].
An unstable particle called the pion has a mean lifetime of 25 ns in its own rest frame. A beam of pions travels through the laboratory at a speed of 0.60c. (a) What is the mean life time of the
Octavio, traveling at a speed of 0.60c, passes Tracy and her barn. Tracy, who is at rest with respect to her barn, says that the barn is 16 m long in the direction in which Octavio is traveling, 4.5
A spaceship resting on Earth has a length of 35.2 m. As it departs on a trip to another planet, it has a length of 30.5 m as measured by the Earth bound observers. The Earthbound observers also
At the 10.0-km-long Stanford Linear Accelerator, electrons with rest energy of 0.511 MeV have been accelerated to a total energy of 46 GeV. How long is the accelerator as measured in the reference
Consider the following decay process: π+ → μ+ + v. The mass of the pion (π+) is 139.6 MeV/c2, the mass of the muon (μ+) is 105.7 MeV/c2, and the mass of the neutrino (v) is negligible. If the
A neutron (mass 1.008 66 u) disintegrates into a proton (mass 1.007 28 u), an electron (mass 0.000 55 u), and an antineutrino (mass 0). What is the sum of the kinetic energies of the particles
A starship takes 3.0 days to travel between two distant space stations according to its own clocks. Instruments on one of the space stations indicate that the trip took 4.0 days. How fast did the
Two spaceships are observed from Earth to be approaching each other along a straight line. Ship A moves at 0.40c relative to the Earth observer, and ship B moves at 0.50c relative to the same
A neutron, with rest energy 939.6 MeV, has momentum 935 MeV/c downward. What is its total energy?
Suppose that as you travel away from Earth in a spaceship, you observe another ship pass you heading in the same direction and measure its speed to be 0.50c. As you look back at Earth, you measure
(a) If you measure the ship that passes you in Problem 68 to be 24 m long, how long will the observers on Earth measure that ship to be? (b) If there is a rod on your spaceship that you measure to
Suppose your handheld calculator will show six places beyond the decimal point. How fast (in meters per second) would an object have to be traveling so that the decimal places in the value of γ can
Verify that the collision between the proton and the nitrogen nucleus in Example 26.4 is elastic.
Muons are created by cosmic-ray collisions at an elevation h (as measured in Earth's frame of reference) above Earth's surface and travel downward with a constant speed of 0.990c. During a time
Refer to Example 26.1. Ashlin travels at speed 0.800c to a star 30.0 ly from Earth. (a) Find the distance between Earth and the star in the astronaut's frame of reference. (b) How long (as measured
An extremely relativistic particle is one whose kinetic energy is much larger than its rest energy. Show that for an extremely relativistic particle E ≈ pc.
A spaceship is moving away from Earth with a constant velocity of 0.80c with respect to Earth. The spaceship and an Earth station synchronize their clocks, setting both to zero, at an instant when
Refer to Example 26.2. One million muons are moving toward the ground at speed 0.9950c from an altitude of 4500 m. In the frame of reference of an observer on the ground, what are? (a) The distance
A charged particle is observed to have a total energy of 0.638 MeV when it is moving at 0.600c. If this particle enters a linear accelerator and its speed is increased to 0.980c, what is the new
A spaceship is traveling away from Earth at 0.87c. The astronauts report home by radio every 12 h (by their own clocks). (a) At what interval are the reports sent to Earth, according to Earth clocks?
A spaceship is traveling away from Earth at 0.87c. The astronauts report home by radio every 12 h (by their own clocks). At what interval are the reports sent to Earth, according to Earth clocks?
A source and receiver of EM waves move relative to one another at velocity v. Let v be positive if the receiver and source are moving apart from one another. The source emits an EM wave at frequency
You are trying to communicate with a rocket ship that is traveling at 1.2 × 108 m/s away from Earth on its way into space. If you send a message at a frequency of 55 kHz, to what frequency should
An astronaut has spent a long time in the Space Shuttle traveling at 7.860 km/s. When he returns to Earth, he is 1.0 s younger than his twin brother. How long was he on the shuttle?
Two atomic clocks are synchronized. One is put aboard a spaceship that leaves Earth at t = 0 at a speed of 0.750c. (a) When the spaceship has been traveling for 48.0 h (according to the atomic clock
A spaceship travels at constant velocity from Earth to a point 710 ly away as measured in Earth's rest frame. The ship's speed relative to Earth is 0.9999c. A passenger is 20 yr old when departing
Convert the following to units in which c = 1, expressing everything in terms of m and kg: (a) Worked example: 10 J. In SI units, 10 J = 10 kgm2 s−2. Since c = 1, we have 1 s = 3 × 108 m, and so 1
(a) The Einstein velocity-addition law, Eq. (1.13), has a simpler form if we introduce the concept of the velocity parameter u, defined by the equationv = tanh u.Notice that for −∞v = tanh uAndw
(a) Using the velocity parameter introduced in Exer. 18, show that the Lorentz transformation equations, Eq. (1.12), can be put in the form t̅ = t cosh u − x sinh u, y̅ = y, x̅ = −t sinh u + x
Convert the following from natural units (c = 1) to SI units: (a) A velocity v = 10−2. (b) Pressure 1019 kgm−3. (c) Time t = 1018 m. (d) Energy density u = 1kgm−3. (e) Acceleration 10m−1.
Write the Lorentz transformation equations in matrix form?
Show that if two events are time like separated, there is a Lorentz frame in which they occur at the same point, i.e. at the same spatial coordinate values?
Show that Eq. (1.2) contains only Mαβ + Mβα when α ( β, not Mαβ and Mβα independently. Argue that this enables us to set Mαβ = Mβα without loss of generality?
(a) Derive Eq. (1.3) from Eq. (1.2), for general {Mαβ, α, β = 0, . . . , 3}. (b) Since (s̅2 = 0 in Eq. (1.3) for any {(xi}, replace (xi by − (xi in Eq. (1.3) and subtract the resulting
Prove Eq. (2.13) from the equation
The following matrix gives a Lorentz transformation from O to:(a) What is the velocity (speed and direction) of relative to O? (b) What is the inverse matrix to the given one? (c) Find the
(a) Compute the four-velocity components in O of a particle whose speed in O is v in the positive x direction, by using the Lorentz transformation from the rest frame of the particle. (b) Generalize
Derive the Einstein velocity-addition formula by performing a Lorentz transformation with velocity v on the four-velocity of a particle whose speed in the original frame was W.
(a) Prove that any time like vector for which U0 > 0 and · = −1 is the four-velocity of some world line. (b) Use this to prove that for any time like vector there is a Lorentz frame in
(a) Show that the sum of any two orthogonal spaces like vectors is space like. (b) Show that a time like vector and a null vector cannot be orthogonal.
Identify the free and dummy indices in the following equations and change them into equivalent expressions with different indices. How many different equations does each expression represent? (a)
The world line of a particle is described by the equations x(t) = at + b sin ωt, y(t) = b cos ωt, z(t) = 0, |bω| < 1, In some inertial frame describe the motion and compute the components of the
A particle of rest mass m has three-velocity v. Find its energy correct to terms of order |v|4. At what speed |v| does the absolute value of 0(|v|4) term equal ½ of the kinetic-energy term 1/2m|v|2?
Prove that conservation of four-momentum forbids a reaction in which an electron and positron annihilate and produce a single photon (γ -ray). Prove that the production of two photons is not
Two identical bodies of mass 10 kg are at rest at the same temperature. One of them is heated by the addition of 100 J of heat. Both are then subjected to the same force. Which accelerates faster,
Prove, using the component expressions, Eqs. (2.24) and (2.26), that
A photon of frequency ν is reflected without change of frequency from a mirror, with an angle of incidence θ. Calculate the momentum transferred to the mirror. What momentum would be transferred if
Let a particle of charge e and rest mass m, initially at rest in the laboratory; scatter a photon of initial frequency νi. This is called Compton scattering. Suppose the scattered photon comes off
Space is filled with cosmic rays (high-energy protons) and the cosmic microwave background radiation. These can Compton scatter off one another. Suppose a photon of energy hν = 2 × 10−4 eV
Show that, if ,, and are any vectors and α and β any real numbers, (α)∙ = α(∙) ∙(β) = β(∙) ∙(+ ) = ∙ + ∙ (+) ∙ = ∙C + ∙
A collection of vectorsIs said to be linearly independent if no linear combination of them is zero except the trivial one, (a) Show that the basis vectors in Eq. (2.9) are linearly independent. (b)
In the t − x spacetime diagram of O, draw the basis vectors 0 and 1. Draw the corresponding basis vectors of the frame that moves with speed 0.6 in the positive x direction relative to O. Draw
(a) Prove that the zero vectors (0, 0, 0, 0) has these same components in all reference frames. (b) Use (a) to prove that if two vectors have equal components in one frame, they have equal components
Prove, by writing out all the terms, thatSince the order of summation doesn't matter, we are justified in using the Einstein summation convention to write simply Which doesn't specify the order of
(a) Given an arbitrary set of numbers {MÎ±Î² ; Î± = 0, . . . , 3; Î² = 0, . . . , 3} and two arbitrary sets of vector components
(a) Given a frame O whose coordinates are {xα}, show that ∂xα / ∂xβ = δαβ. (b) For any two frames, we have, ∂xβ / ∂xa̅ = Λβa̅. Show that (a) and the chain rule imply Λ(a̅ Λ
Use the notation ˆ‚Ï† / ˆ‚xÎ± = Ï†,Î± to re-write Eqs. (3.14), (3.15), and (3.18).ˆ‚xÎ² /
Let S be the two-dimensional plane x = 0 in three-dimensional Euclidean space. Let ( 0 be a normal one-form to S.(a) Show that if is a vector which is not tangent to S, then () ( 0.(b) Show that
Prove, by geometric or algebraic arguments, that f is normal to surfaces of constant f?
Let → O (1, 1, 0, 0) and → O (- 1, 0, 1, 0) be two one forms. Prove, by trying two vectors and as arguments, that ( ( ( . Then find the components of ( ?
Supply the reasoning leading from Eq. (3.23) to Eq. (3.24)?f = fa(a(; a((e ⃗μ e ⃗v) = (aμ ((v,
(a) Prove that h(s) defined byIs an symmetric tensor.(b) Prove that h(A) defined byIs an antisymmetric tensor.(c) Find the components of the symmetric and antisymmetric parts of Š— defined in
(a) Find the one-forms mapped by the metric tensor from the vectors→ O (1, 0, - 1, 0)→ O (1, 0, - 1, 0)→ O (-1, 0, - 1, 0)→ O(0, 0, 0, 1, 1).(b) Find the vectors mapped by the inverse of
(a) Prove that the matrix {ηαβ} is inverse to {ηαβ} by performing the matrix multiplication.(b) Derive Eq. (3.53).( : = ½ [( + )2 - 2 - 2].
To prove that the set of all one-forms is a vector space, we must show that this set meets the axioms (1) and (2) given in Appendix A, p. 374?
a) Let a region of the t − x plane be bounded by the lines t = 0, t = 1, x = 0, x = 1. Within the t − x plane, find the unit outward normal one-forms and their associated vectors for each of the
Suppose that instead of defining vectors first, we had begun by defining one-forms, aided by pictures like Fig. 3.4. Then we could have introduced vectors as linear realvalued functions of one-forms,
(a) Given the components of a (20) tensor MÎ±Î² as the matrixfind: (i) The components of the symmetric tensor M(Î±Î²) and the antisymmetric tensor
Show that if A is a (20) tensor and B a (02) tensor, thenAa(Ba(?
Suppose A is an antisymmetric (20) tensor, B a symmetric (02) tensor, C an arbitrary (02) tensor, and D an arbitrary (20) tensor, Prove?(a) Aa( Ba( = 0;(b) Aa( Ca( = Aa(C[a(];(c) Ba(Da( = Ba(D(a().
(a) Suppose A is an antisymmetric (20) Show that {Aa(}, obtained by lowering indices by using the metric tensor, are components of an antisymmetric (02) tensor?(b) Suppose Va = Wa. Prove that Va = Wa?
Deduce Eq. (3.66) from Eq. (3.65). (T: = (Ta(,(((((a) dT / dr = (Ta(,(((a)U(?
Prove that tensor differentiation obeys the Leibniz (product) rule? ((A ( B) = ((A) ( B + A ( ((B).
(a) Prove, by writing out all the terms, the validity of the following
Consider a time like unit four-vector , and the tensor P whose components are given byPÎ¼v = Î·Î¼v + UÎ¼Uv.(a) Show that P is a projection operator that projects an arbitrary vector into one
Given the following vectors in :(a) Show that they are linearly independent;(b) Find the components of if(c) Find the value of () for(d) Determine whether the one-forms and , , , and are linearly
Justify each step leading from Eqs. (3.10a) to (3.10d)? Aa̅ pa̅ = (Λa̅βAβ) (Λμa̅pμ) = Λμa̅ Λa̅ βAβpμ, = δμβAβpμ, = Aβpβ.
Consider the basis {a} of a frame O and basisFor the space of one forms, where we have {(} is not the basis dual to {}. (a) Show that ( (a) a for arbitrary . (b) Let †’ O (1, 1, 1, 1). Find
Draw the basis one forms t and x of a frame O?
Fig. 3.5 shows curves of equal temperature T (isotherms) of a metal plate. At the points P and Q as shown, estimate the components of the gradient T?Isotherms of an irregularly heated plate.
Comment on whether the continuum approximation is likely to apply to the following physical systems: (a) Planetary motions in the solar system; (b) Lava flow from a volcano; (c) Traffic on a major
Take the limit of Eq. (4.35) for |v|
(a) Show that the matrix δij is unchanged when transformed by a rotation of the spatial axes. (b) Show that any matrix which has this property is a multiple of δij.
In the MCRF, Ui = 0. Why can't we assume Ui,( = 0?
We have defined (( = U(,( U(. Go to the nonrelativistic limit (small velocity) and show that (i = vi + (v ( ()vi = Dvi / Dt, Where the operator D/Dt is the usual 'total' or 'advective' time
Sharpen the discussion at the end of 4.6 by showing that -(p is actually the net force per unit volume on the fluid element in the MCRF.
Show that Eq. (4.58) can be used to prove Gauss' law, Eq. (4.57).In Eq. (4.58)In Eq. (4.57)
Flux across a surface of constant x is often loosely called 'flux in the x direction'. Use your understanding of vectors and one-forms to argue that this is an inappropriate way of referring to a
(a) Describe how the Galilean concept of momentum is frame dependent in a manner in which the relativistic concept is not. (b) How is this possible, since the relativistic definition is nearly the

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