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NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET INSTITUTT FOR FYSIKK

Kontakt under eksamen: Jon Andreas Støvneng Telefon: 73 59 36 63 / 45 45 55 33

EXAM TFY4160 BØLGEFYSIKK Mandag 3. desember 2007 kl. 0900 - 1300 English version

Hjelpemiddel: C

• K. Rottmann: Matematisk formelsamling (alle språk).

• O. Øgrim og B. E. Lian: Størrelser og enheter i fysikk og teknikk, eller C. Angell og B. E. Lian: Fysiske størrelser og enheter.

• Typegodkjend kalkulator, med tomt minne, i henhold til liste utarbeidd av NTNU. (HP30S eller liknende.)

Page 2 - 6: Exercises. Page 7 - 15: Formulas.

There are 7 questions. The weight of each question is given. Vectors are given in bold letters. Unit vectors are given with a hat above the symbol.

The marks are ready in early January.

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OPPGAVE 1 [Counts 20%]

A mass m is connected to a spring with spring constant k. The damping force is proportional to the speed, −bẋ. An external force F_{0} sin ωt forces the mass to oscillate back and forth with angular frequency ω.

• Show that the motion of the mass is determined by the equation

ẍ + 2δẋ + ω_{0}^{2}x = A_{0} sin ωt,

and thus determine δ, ω_{0} and A_{0}. Here, x is the displacement of the mass relative to its equilibrium position x = 0.

• The solution of this equation is

x(t) = x_{0}(ω) cos(ωt + ϕ)

where the frequency dependence of the amplitude is given by

x_{0}(ω) = \frac{A_{0}}{\sqrt{(ω_{0}^{2} − ω^{2})^{2} + (2ωδ)^{2}}}

Assume weak damping, i.e. δ ≪ ω_{0}, and find the full width at half maximum, Δω, of the resonance peak, i.e., the difference between the two angular frequencies ω_{±} = ω_{0} ± Δω/2 that both result in an amplitude x_{0} equal to the maximum amplitude divided by \sqrt{2}. (With weak damping, maximum amplitude is obtained for ω ≃ ω_{0}.)

—————–

Now, the speed dependent damping is replaced by ordinary friction against a surface with friction coefficient μ (the same both static and dynamic), so that there is a frictional force μmg, with direction opposite to the motion of the mass. Here, g is the acceleration due to gravity. (If the mass is at rest, the frictional force is up to μmg, with direction opposite to other forces trying to make the mass move.) We also turn off the external force and study free oscillations. The figure shows, as an example, the situation with x > 0 and ẋ > 0.

Position x = 0 for the mass corresponds to a spring which is not stretched or compressed. At time t = 0, the mass is pulled to the right, to the position x(0) = 4.5μmg/k, and released with zero initial velocity.

• Show that the motion of the mass is described by the equation(s)

ξ̈+ ω^{2}ξ = 0

where we have introduced ξ = x ± μmg/k, and where the sign depends on whether the mass moves to the right or to the left. What is the angular frequency ω?

• Sketch x(t) from t = 0 to t = 3T/2. Here, T = 2π/ω is the period of the oscillation. Give a short comment to the graph.

OPPGAVE 2 [Counts 15%]

In this exercise, we study surface waves on deep water. These waves have the dispersion relation

ω(k) = \sqrt{gk + γk^{3}/ρ}

Here, k is the wave number, g = 9.8 m/s^{2} the acceleration due to gravity, ρ = 10^{3} kg/m^{3} the mass density of water, and γ = 0.073 N/m the surface tension of water.

• Find the period of waves with wave length λ = 2.0 m.

• Find the wave lengths of waves with phase velocity v_{f} = 1.0 m/s.

• A wave packet has 9 clear crests and wave length 2.0 m. We assume that the total length of the wave packet is 8.5 wave lengths, i.e. 17.0 m, and that the total length does not change much during our observations. The figure shows a snapshot of the wave packet:

We focus on a fixed position x_{0} and watch carefully as the wave packet passes by. How long time does the wave packet spend in passing x_{0}? How many crests do we observe passing x_{0}? Explain your answer. (Comment: Such a spatially restricted wave packet ”contains” more wave lengths than the mentioned one of 2.0 m. However, the information that the length of the packet does not change during the observations, is sufficient to answer the questions.)

OPPGAVE 3 [Counts 15%]

Monochromatic light with wave length λ = 632.8 nm falls perpendicularly on 4 narrow parallel slits. Each slit has width a = 40.00 μm, and the center-to-center distance between the slits is d = 125.0 μm. The light that passes through the slits falls on a screen located at a distance of L = 300.0 cm from the slits, and results in an intensity pattern I(y). In the figure, the length direction of the slits is perpendicular to the paper plane:

• Write down an expression for the intensity pattern I(y) which is valid for small angles of deviation θ. In this expression, only y should remain as variable parameter (in addition to the prefactor I_{0}, which represents 1/16 of the intensity in y = 0). Use 4 digits for numerical factors in your expression, and state the unit you have chosen for y (m, cm, mm ..).

• How many zeros does I(y) have all together inside the central diffraction maximum? (By ”central diffraction maximum” is meant the region between y_{0} and −y_{0}, where y_{0} corresponds to the position of the first zero if we had only one slit of width a.) Sketch I(y) for −y_{0} < y < y_{0}.

OPPGAVE 4 [Counts 10%]

Unpolarized white light is directed towards a tank containing a scattering medium. The scatterers in the tank may be regarded as induced oscillating electric dipoles with dipole moment p in the same direction as E of the incident light. (The concentration of scatterers is so small that we may disregard the possibility of scattered light being scattered again.) The incident light passes through an ideal polarizer, so that the light striking the tank has E ∥ ŷ.

• Discuss qualitatively (and briefly) the intensity, the polarization, and the color of the light being scattered in both the y and the z direction.

OPPGAVE 5 [Counts 10%]

A monochromatic plane electromagnetic wave falls on a plane boundary between two nonmagnetic dielectric media 1 and 2, see figure above. If the wave is polarized parallel to the plane of incidence, the relation between the amplitude of the reflected wave, E_{rp0}, and the amplitude of the incident wave, E_{ip0}, is

E_{rp0} = \frac{α − β}{α + β} E_{ip0}

Here, α ≡ cos θ_{2}/ cos θ_{1} and β ≡ μ_{1}v_{1}/μ_{2}v_{2}, where μ_{j} and v_{j} are magnetic permeability and speed velocity, respectively, in medium nr j (j = 1, 2). With nonmagnetic media, we have μ_{1} = μ_{2} = μ_{0}. The incident wave is completely transmitted if the angle of incidence θ_{1} equals the Brewster angle θ_{B}.

• Show that the Brewster angle is given by

tan θ_{B} = \frac{n_{2}}{n_{1}} ,

where n_{1} and n_{2} are index of refraction in medium 1 and 2, respectively.

OPPGAVE 6 [Counts 10%]

• Show that an electromagnetic wave in vacuum, with

E(r, t) = E_{0} sin(k · r − ωt)

and

B(r, t) = B_{0} sin(k · r − ωt)

is transversal. Hint: Maxwell’s equations.

OPPGAVE 7 [Counts 20%]

Two equal masses m move straight towards each other. Before they collide, the masses 1 and 2 have velocity c/2 and −c/2, respectively, as measured in the lab system S_{3}. Here, c is the speed of light. (In other words, we have chosen positive velocities to the right.) The masses 1 and 2 are, prior to collision, at rest in the inertial frames S_{1} and S_{2}, respectively.

• Before the collision, the masses 1 and 2 are spherically shaped in their rest frames S_{1} and S_{2}, respectively. What is their shape in S_{3}? Explain your answer.

• Let us with v_{ij} denote the velocity of S_{i} as observed in S_{j} (i, j = 1, 2, 3), so that v_{13} = c/2 and v_{23} = −c/2. Determine the other relative velocities v_{31}, v_{32}, v_{12}, and v_{21}.

• What is the total (relativistic) momentum p_{3} and energy E_{3} of this system in S_{3}? (With ”system”, we simply mean the two masses 1 and 2.) Also determine the total momentum p_{1} and energy E_{1} in S_{1}.

The two masses collide, completely inelastically, and stick together as a single body with mass M after the collision.

• Use the principles of momentum and energy conservation to determine M. What is the velocity v_{1} of this body as measured in S_{1}? (S_{1} is still the inertial reference frame moving with velocity c/2 relative to the lab system S_{3}.)

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Formulas

Bold symbols denote vectors. Symbols with hat above are unit vectors. Validity of formulas and meaning of symbols are assumed to be known.

• Harmonic plane wave:

ξ(x, t) = ξ_{0} sin(kx − ωt + ϕ)
ξ(r, t) = ξ_{0} sin(k · r − ωt + ϕ)

• Wave equation:

\frac{∂^{2}ξ(x, t)}{∂x^{2}} = \frac{1}{v^{2}} \frac{∂^{2}ξ(x, t)}{∂t^{2}}

∇^{2}ξ(r, t) (≡ \frac{∂^{2}ξ}{∂x^{2}} + \frac{∂^{2}ξ}{∂y^{2}} + \frac{∂^{2}ξ}{∂z^{2}} ) = \frac{1}{v^{2}} \frac{∂^{2}ξ(r, t)}{∂t^{2}}

• Phase velocity:

v = \frac{ω}{k}

• Group velocity:

v_{g} = \frac{dω}{dk}

• General, for nondispersive undamped waves: v = \sqrt{\frac{elastic modulus}{massdensity}}

• General, for linear response in elastic media: mechanical stress = elastic modulus × relative strain

• For transverse waves on string:

v = \sqrt{\frac{S}{μ}}

• For longitudinal waves in fluids:

v = \sqrt{\frac{B}{ρ}}

• For longitudinal waves in solids:

v = \sqrt{\frac{Y}{ρ}}

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• Average of harmonically varying A(x, t), averaged over wave length λ:

\overline{A} = \frac{∫_{0}^{λ} A(x, t) dx}{∫_{0}^{λ} dx} = \frac{1}{λ} ∫_{0}^{λ} A(x, t) dx

Average of harmonically varying A(x, t), averaged over period T:

⟨A⟩ = \frac{∫_{0}^{T} A(x, t) dt}{∫_{0}^{T} dt} = \frac{1}{T} ∫_{0}^{T} A(x, t) dt

• Average energy pr unit length for harmonic wave on string:

\overline{ε} = \frac{1}{2}μω^{2}ξ_{0}^{2}

• Average energy pr unit volume for harmonic plane wave:

\overline{ε} = \frac{1}{2}ρω^{2}ξ_{0}^{2}

• Average power transported with harmonic wave on string:

\overline{P} = v\overline{ε} = \frac{1}{2}vμω^{2}ξ_{0}^{2}

• Average intensity in harmonic plane wave:

I = v\overline{ε} = \frac{1}{2}vρω^{2}ξ_{0}^{2}

• Average momentum density for harmonic wave:

\overline{π} = \frac{\overline{ε}}{v}

• Ideal gas:

pV = Nk_{B}T

• Heat capacity at constant pressure (Q = heat):

C_{p} = (\frac{dQ}{dT})_{p}

• Heat capacity at constant volume (Q = heat):

C_{V} = (\frac{dQ}{dT})_{V}

• Adiabatic conditions (i.e. no heat exchange): pV^{γ} = constant

• The adiabatic constant:

γ = \frac{C_{p}}{C_{V}}

Gas with 1-atomic molecules: γ = 5/3. Gas with 2-atomic molecules: γ = 7/5.

• Bulk modulus for ideal gas at adiabatic conditions: B = γp

• Sound velocity in gas (m = molecular mass):

v = \sqrt{\frac{γp}{ρ}} = \sqrt{\frac{γk_{B}T}{m}}

• Sound pressure:

Δp = −B \frac{∂ξ}{∂x}

• Sound level:

β(dB) = 10 log \frac{I}{I_{0}}

with I_{0} = 10^{−12} W/m^{2}

• Doppler effect:

ν_{O} = \frac{1 − v_{O}/v}{1 − v_{S}/v} ν_{S}

• For shock waves:

sin α = \frac{v}{v_{S}}

• Transverse wave on string with mass density μ_{1} for x < 0 and μ_{2} for x > 0, incident wave propagating in positive x direction:

Amplitude for reflected wave:

y_{r0} = \frac{\sqrt{μ_{2}} − \sqrt{μ_{1}}}{\sqrt{μ_{2}} + \sqrt{μ_{1}}} y_{i0}

Amplitude for transmitted wave:

y_{t0} = \frac{2\sqrt{μ_{1}}}{\sqrt{μ_{2}} + \sqrt{μ_{1}}} y_{i0}

Reflection coefficient:

R = \begin{matrix}\overline{P} & r \\ \overline{P} & i \end{matrix}

Transmission coefficient:

T = \begin{matrix}\overline{P} & t \\ \overline{P} & i \end{matrix}

• Plane sound wave with normal incidence on boundary in x = 0 between two media with elastic moduli and mass densities E_{1}, ρ_{1} (for x < 0) and E_{2}, ρ_{2} (for x > 0), respectively. Incident wave propagating in positive x direction:

Amplitude for reflected wave:

ξ_{r0} = \frac{\sqrt{ρ_{2}E_{2}} − \sqrt{ρ_{1}E_{1}}}{\sqrt{ρ_{2}E_{2}} + \sqrt{ρ_{1}E_{1}}} ξ_{i0}

Amplitude for transmitted wave:

ξ_{t0} = \frac{2\sqrt{ρ_{1}E_{1}}}{\sqrt{ρ_{2}E_{2}} + \sqrt{ρ_{1}E_{1}}} ξ_{i0}

Reflection coefficient:

R = \begin{matrix}\overline{P} & r \\ \overline{P} & i \end{matrix}

Transmission coefficient:

T = \begin{matrix}\overline{P} & t \\ \overline{P} & i \end{matrix}

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• Maxwell’s equations on integral form:

∮ E · dA = q/ε_{0}
∮ B · dA = 0

∮ E · dl = −\frac{d}{dt} ∫ B · dA

∮ B · dl = μ_{0}I + μ_{0}ε_{0} \frac{d}{dt} ∫ E · dA

• Maxwell’s equations on differential form: ∇ · E = ρ/ε_{0} ∇ · B = 0

∇ × E = −\frac{∂B}{∂t}

∇ × B = μ_{0}j + μ_{0}ε_{0} \frac{∂E}{∂t}

• Lorentz force:

F = q (E + v × B)

• Wave equation for E and B in vacuum:

∇^{2}E = \frac{1}{c^{2}} \frac{∂^{2}E}{∂t^{2}}

∇^{2}B = \frac{1}{c^{2}} \frac{∂^{2}B}{∂t^{2}}

c = 1/\sqrt{ε_{0}μ_{0}}

• Energy density in electromagnetic field:

u = u_{E} + u_{B} = \frac{1}{2}ε_{0}E^{2} + \frac{1}{2μ_{0}} B^{2}

• Intensity in electromagnetic wave:

I = cε_{0}\overline{E^{2}} = cε_{0}⟨E^{2}⟩

• Poynting’s vector:

S = \frac{1}{μ_{0}} E × B

• Momentum in electromagnetic wave:

π = μ_{0}ε_{0}S

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• Electric dipole moment:

p = qd

• Magnetic dipole moment:

m = IA

• Average radiated effect from oscillating electric dipole p_{0} cos(ωt):

⟨P ⟩ = \frac{p_{0}^{2}ω^{4}}{12πε_{0}c^{3}}

• Average radiated effect from oscillating magnetic dipole m_{0} cos(ωt):

⟨P ⟩ = \frac{μ_{0}m_{0}^{2}ω^{4}}{12πc^{3}}

• Malus’ law:

I(θ) = I_{0} cos^{2} θ

• Linear media:

P = ε_{0}χ_{e}E
D = ε_{0}E + P = ε_{0}(1 + χ_{e})E = ε_{0}ε_{r}E = εE
M = χ_{m}H
B = μ_{0}H + M = μ_{0}(1 + χ_{m})H = μ_{0}μ_{r}H = μH

• Maxwell’s equations etc:

∮ D · dA = q_{free}
∮ B · dA = 0
∮ E · dl = −\frac{d}{dt} ∫ B · dA
∮ H · dl = I_{free} + \frac{d}{dt} ∫ D · dA
∇ · D = ρ_{free}
∇ · B = 0
∇ × E = −\frac{∂B}{∂t}
∇ × H = j_{free} + \frac{∂D}{∂t}

• Energy density, Poynting’s vector:

u = \frac{1}{2}εE^{2} + \frac{1}{2μ}B^{2}
S = \frac{1}{μ}E × B

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• For electromagnetic waves in media (q_{free} = I_{free} = 0):

∇^{2}E = \frac{1}{v^{2}} \frac{∂^{2}E}{∂t^{2}}
∇^{2}B = \frac{1}{v^{2}} \frac{∂^{2}B}{∂t^{2}}
v = \frac{1}{\sqrt{εμ}} = \frac{c}{\sqrt{ε_{r}μ_{r}}} = \frac{c}{n}

• Boundary conditions (q_{free} = I_{free} = 0 in the boundary):

ΔD_{⊥} = 0
ΔE_{∥} = 0
ΔB_{⊥} = 0
ΔH_{∥} = 0

• Reflection and refraction:

θ_{r} = θ_{i}
n_{1} sin θ_{i} = n_{2} sin θ_{t}

• Young’s experiment with two narrow slits:

I(θ) = 4I_{0} cos^{2} (\frac{πd}{λ} sin θ)

• Diffraction grating with N narrow slits:

I(θ) = I_{0} sin^{2} (\frac{Nπd}{λ} sin θ) ; sin^{2} (\frac{πd}{λ} sin θ)

• Diffraction from one slit:

I(θ) = I(0) \begin{matrix}sin & 2 & (\frac{πa}{λ} sin θ) \\ (\frac{πa}{λ} sin θ) & 2 \end{matrix}

• Diffraction from N slits of width a:

I(θ) = I_{0} sin^{2} (\frac{πa}{λ} sin θ) ; (\frac{πa}{λ} sin θ)^{2} sin^{2} (\frac{Nπd}{λ} sin θ) ; sin^{2} (\frac{πd}{λ} sin θ)

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• Lorentz factor:

γ = (1 − v^{2}/c^{2})^{−1/2}

• Lorentz transformations (\overline{S} has velocity v = vx̂ relative to S):

\overline{x} = γ (x − vt)
\overline{y} = y
\overline{z} = z
\overline{t} = γ (t − \frac{v}{c^{2}} x)
x = γ (\overline{x} + v\overline{t})
y = \overline{y}
z = \overline{z}
t = γ (\overline{t} + \frac{v}{c^{2}} \overline{x})

• Time dilation:

Δt = γΔ\overline{t}

• Length contraction:

Δ\overline{x} = γΔx

• Velocity in S (u = u_{x}x̂ + u_{y}ŷ + u_{z}ẑ):

u_{x} = dx/dt
u_{y} = dy/dt
u_{z} = dz/dt

Velocity in \overline{S} (\overline{u} = \overline{u}_{x}x̂ + \overline{u}_{y}ŷ + \overline{u}_{z}ẑ):

\overline{u}_{x} = d\overline{x}/d\overline{t}
\overline{u}_{y} = d\overline{y}/d\overline{t}
\overline{u}_{z} = d\overline{z}/d\overline{t}

• Addition of velocities (all velocities in the same direction):

v_{AC} = \frac{v_{AB} + v_{BC}}{1 + v_{AB}v_{BC}/c^{2}}

• Doppler effect for electromagnetic waves:

\overline{ν} = ν (\frac{c − v}{c + v} )^{1/2}

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• Relativistic momentum:

p = γmv

• Newton’s 2. law:

F = \frac{dp}{dt}

• Energy:

E = γmc^{2}
E_{0} = mc^{2}
E_{k} = E − E_{0}
E^{2} = (pc)^{2} + (mc^{2})^{2}

• Elastic process: E, p, E_{k} and m conserved.

• Inelastic process: E and p conserved.