Questions and Answers of The Physics Energy

Consider an amount of helium gas at atmospheric pressure and room temperature (T = 300K) enclosed in a 1-liter partition within a cubic meter. The volume outside the partition containing
What is the information entropy of the results of spinning a roulette wheel with 38 possible equally likely outcomes (36 numbers, 0 and 00) 10 times? How many fair coins would you need to flip to get
What is the information entropy of the results of flipping a biased coin 1000 times, if the coin comes up tails with probability 5/6 and heads with probability 1/6? For this weighted coin, can you
The polynomials appearing in eqs. are known as Hermite polynomials. Compute the second, third, and fourth excited wave functions by assuming quadratic, cubic, and quartic polynomials and
The energy required to remove an electron from an atom is called the ionization energy. Predict the energy needed to ionize the last electron from an atom with atomic number Z. (The atomic number is
Find the ground state of the quantum harmonic oscillator. Start by showing that the Hamiltonian from eq can be ?factorized? similar to (a2 ? b2) = (a ? b)(a + b) in the form Where? Show that
Show that the usual relation E = p2/2m arises as the leading momentum dependence of the energy in an expansion of the relativistic p relation? for small p ? mc. Vm²cª + p?c² E 24 C* + p²c2
Show that the solution to Schr?dinger?s equation for a three-dimensional harmonic oscillator is? where ?n(x) is a solution to the one-dimensional oscillator with energy (n + 1/2)??. Show that the
Consider a macroscopic oscillator, given by a mass of m = 0.1 kg on a spring with spring constant k = 0.4N/m. What is the natural frequency ω of this oscillator? Considering this as a quantum
Consider a system of two independent harmonic oscillators, each with natural angular frequency ω. We denote states of this system by |nm〉 where the nonnegative integers n,m denote the excitation
The interatomic potential for diatomic molecules like O2 and N2 can be approximated near its minimum by a simple harmonic oscillator potential 1/2 kx2. Consider the classical motion of a diatomic
A particle with spin S has magnetic moment m = qgS/2Mc (see Example 7.3). Solving for the classical motion of S in a constant magnetic field B, we found that S precesses around B with angular
Show that the set of functions ψk(x) = eikx (for real k) are the solutions to the condition ψ(x+δ) = eiθ(δ)ψ(x) (for real θ). use sequential translations by δ1 and δ2 to show that θ
In the text we introduced electron spin states with spin ±1/2 in the ẑ-direction, |±〉, and states with spin ±1/2 in the x̂-direction, |±x〉 = (|+〉 ± |−〉)/√2. Measuring the
A quantum particle is restricted to a one dimensional box 0 ? x ? L. It experiences no forces within the box, but cannot escape. At time t = 0, the particle is in the state ? where ?i(x) are the
A particle of mass m in a potential V(x) has an energy basis state with wave function ? What is the potential and what is the energy of this state? What is the value of the constant C? Y(x) = C/
Suppose an electron, constrained to move only in the vertical direction, sits on an impenetrable table in Earth’s gravitational field. Write down the (one dimensional) Schrödinger equation that
Nuclear forces are so strong that they keep protons and neutrons in a spherical region a few femtometers in radius (1 fm = 10−15 m). To get a crude idea of the energy scales involved in nuclear
In one dimension, compare the plane wave solutions to the heat equation ?T/?t = a?2T/?x2 with plane wave solutions to the free Schr?dinger equation Show that the heat equation has real solutions
1 Planck’s constant is very small on the scale of human affairs. Compare the angular momentum of a child’s marble spinning at 60 rpm with Planck’s constant. Compare the energy of the same
When an object is in radiative equilibrium with its environment at temperature T, the rates at which it emits and absorbs radiant energy must be equal. Each is given by dQ0/dt = εσT4A. If the
Consider a region with average surface temperature T0 = 10 ◦C, annual fluctuations of ΔT = 30 ◦C, and surface soil with a ≅ 2.4 × 10−7 m2/s, k ≅ 0.3W/mK. If the local upward heat flux
In areas where the soil routinely freezes it is essential that building foundations, conduits, and the like be buried below the frost level. Central Minnesota is a cold part of the US with yearly
Ignoring convective heat transfer, estimate the change in U-factor by replacing argon by krypton in the quadruple glazed windows described in Table 6.3. You can ignore radiative heat transfer since
An insulated pipe carries a hot fluid. The setup is shown in Figure 6.18. The copper pipe has radius R0 = 1 cm and carries a liquid at T0 = 100 ◦C. The pipe is encased in a cylindrical layer of
z A building wall is constructed as follows: starting from the inside the materials used are? (a) 1/2" gypsum wallboard;? (b) Wall-cavity, 80% of which is occupied by a 3.5" fiberglass batt and 20%
According to, a double pane window with an emissivity ε = 0.05 coating and a 1/4" air gap has a measured (center of glass) U-factor of 2.33W/m2 K. Assume that this coating is sufficient to stop all
In figure, a film of still air was identified as the source of almost all of the thermal resistance of a single-pane glass window. Determine the thickness of the still air layers on both sides of the
Assume that a 60m2 wall of a house is insulated to an R-value of 5.4 (SI units), but suppose the insulation was omitted from a 1m2 gap where only 6 cm of wood with an R-value of 0.37
Estimate the R-value of the wall? drywall an R36 wall fiber fill R13 with three layers of insulation rigid foam R10 fiberglass R13 sheathing siding I 9 cm 5 cm 9 cm 1 cm 2.5 cm
Consider a building with 3200 ft2 of walls. Assume the ceiling is well-insulated and compute the energy loss through the walls based on the following materials, assuming an indoor temperature of 70
Humans radiate energy at a net rate of roughly 100W; this is essentially waste heat from various chemical processes needed for bodily functioning. Consider four humans in a roughly square hut
Given the Sun’s power output of 384 YW and radius 695500 km, compute its surface temperature assuming it to be a black body with emissivity one.
The heat transfer coefficient h̅ for air flowing at 30m/s over a 1m long flat plate is measured to be 80W/m2 K. Estimate the relative importance of heat transfer by convection and conduction for
Two rigid boards of insulating material, each with area A, have thermal conductances U1 and U2. Suppose they are combined in series to make a single insulator of area A. What is the heat flux across
Suppose a small stoneware kiln with surface area 5m2 sits in a room that is kept at 25 ◦C by ventilation. The 15 cm thick walls of the kiln are made of special ceramic insulation, which has k
Carbon dioxide sublimes at pressures below roughly 5 atm? At a pressure of 2 atm this phase transition occurs at about ?69 ?C with an enthalpy of sublimation of roughly 26 kJ/mol. Suppose a kilogram
Roughly 70% of the 5 × 1014 m2 of Earth’s surface is covered by oceans. How much energy would it take to melt enough of the ice in Greenland and Antarctica to raise sea levels 1 meter? Suppose
A start-up company is marketing steel “ice cubes” to be used in place of ordinary ice cubes. How much would a liter of water, initially at 20 ◦C, be cooled by the addition of 10 cubes of steel,
A new solar thermal plant being constructed in Australia will collect solar energy and store it as thermal energy, which will then be converted to electrical energy. The plant will store some of the
A solar thermal power plant currently under construction will focus solar rays to heat a molten salt working fluid composed of sodium nitrate and potassium nitrate. The molten salt is stored at a
A “low-flow” shower head averages 4.8 L/min. Taking other data from Example 5.3, estimate the energy savings (in J/y) if all the people in your country switched from US code to low-flow shower
How much energy does it take to heat 1 liter of soup from room temperature (20 ◦C) to 65 ◦C?
A cylindrical tube oriented vertically on Earth’s surface, closed at the bottom and open at the top (height 100m, cross-sectional area 1m2) initially contains air at a pressure of one atmosphere
Everyday experience indicates that it is much easier to compress gases than liquids. This property is measured by the isothermal compressibility, β =−1/V ∂V/∂p|T , the fractional change in a
When air is inhaled, its volume remains constant and its pressure increases as it is warmed to body temperature Tbody = 37 ◦C. Assuming that air behaves as an ideal gas and that it is
Non-rigid airships known as blimps have occasionally been used for transportation. A blimp is essentially a balloon of volume V filled with helium. The blimp experiences a buoyancy force F = (ρatm
A cylinder initially contains V0 = 1 L of argon at temperature T0 = 0 ◦C and pressure p0 = 1 atm. Suppose that the argon is somehow made to expand to a final volume V = 2 L in such a way that the
A wave travels to the right on a string with constant tension τ and a mass density that slowly increases from ρ on the far left to ρ′ on the far right. The mass density changes slowly enough
A wave satisfying?eq. passes from one medium in which the phase velocity for all wavelengths is v1 to another medium in which the phase velocity is v2. The incident wave gives rise to a reflected
As stated in the text, the dispersion relation relating the wave number and angular frequency of ocean surface waves is ω = √pgk, where g ≅ 9.8 m/s2. Compute the wavelength and speed of
Consider a cylindrical resistor of cross-sectional area A and length L. Assume that the electric field E and current density j are uniform within the resistor. Prove that the integrated power
What is the pressure exerted by a beam of light on a perfect mirror from which it reflects at normal (perpendicular) incidence? Generalize this to light incident at an angle θ to the normal on an
A string of length L is initially stretched into a “zigzag” profile, with linear segments of string connecting the (x, f(x)) points (0, 0), (L/4, a), (3L/4,−a), (L, 0). Compute the Fourier
A string of length L begins in the configuration y(x) = A[ 1/3 sin k1 x + 2/3 sin k2 x] with no initial velocity. Write the exact time-dependent solution of the string y(x, t). Compute the
Derive eq.??by taking the time derivative of eq.??and using Maxwell?s equations. ди (х, 1) +V. S(x,1) %3 -j(х,1) Е(x, 1) at
Consider two electromagnetic?plane waves eq.? one with amplitude Ea0 and wave vector ka and the other with amplitude Eb0 and wave vector kb. These waves are said to add coherently if the average
It has been proposed that solar collectors could be deployed in space, and that the collected power could be beamed to Earth using microwaves. A potential limiting factor for this technology would be
Suppose an electromagnetic plane wave is absorbed on a surface oriented perpendicular to the direction of propagation of the wave. Show that the pressure exerted by the radiation on the surface is
Derive the wave equation for B analogous to eq.? a2 E 1 aB V x at 1 V x (V x E) at2 HOEO 1 -v²E.
The strongest radio stations in the US broadcast at a power of 50 kW. Assuming that the power is broadcast uniformly over the hemisphere above Earth’s surface, compute the strength of the electric
Compute the maximum energy flux possible for electromagnetic waves in air given the constraint that the electric field cannot exceed the breakdown field.
Show that the energy density on a string u(x, t), defined in eq.? obeys the conservation law?u/?t +?S/?x = 0, where S (x, t) = ??y?y? is the energy flux, the energy per unit time passing a point x.
Derive the equation of motion for the string? from a microscopic model. Assume a simple model of a string as a set of masses ?m spaced evenly on the x axis at regular intervals of ?x, connected by
A violin A-string of length L = 0.33 m with total mass 0.23 g has a fundamental frequency (for the lowest mode) of 440 Hz. Compute the tension on the string. If the string vibrates at the fundamental
Sound waves travel in air at roughly 340 m/s. The human ear can hear frequencies ranging from 20 Hz to 20 000 Hz. Determine the wavelengths of the corresponding sine wave modes and compare to
Take the divergence of both sides of eq. Use Coulomb’s law on the left and current conservation on right to show that the equation is consistent. ӘЕ VхВ - ноєо — 3D Дој. at
Consider the transformer in Figure 3.25 . Suppose the load is a resistor R and that the transformer is ideal, with M2 = LS LP and all magnetic flux lines passing through both inductors. Show that
Design a transmission system to carry power from wind farms in North Dakota to the state of Illinois (about 1200 km). The system should handle Illinois’s summer time electricity demand of 42GW.
Prove that the mutual inductance is a symmetric relation, M12 = M21, by computing the energy stored when current I1 is established in loop 1 and then current I2 is established in loop 2. Then set up
Consider a long, hollow solenoid of volume V.Show that, ignoring end effects, its inductance is L = n2Vμ0, and that the magnetic energy it stores EEM = LI2/2 can be written in the form of eq.
Explain why the integral that appears in eq. is independent of the choice of surface S .[make use of eq. k dS · B dS (V x E) = dt dS B
Derived the EMF on a wire loop rotating in a magnetic field using the Lorentz force law to compute the forces on the mobile charges. Although Faraday?s law of induction does not apply in a rotating
Consider the motor. If the resistance in the wire wrapping the rotor is 1Ω, compute the energy lost under the conditions described. What fraction of energy is lost to Joule heating in this
An electric motor operates at 1000 rpm with an average torque of τ = 0.1Nm. What is its power output? If it is running on 1.2A of current, estimate the back-EMF from the rotor.
Derive eq. (3.52) from eq. (3.43) and eq. (3.50).Make sure you get the both the direction and magnitude. dF (x) = 1 dx x B(x),
Show that the force per unit area on the windings of an air-core solenoid from the magnetic field of the solenoid itself is of order F/A ∼ B2/μ0. Check that the dimensions of this
Review how magnetic fields are calculated from Ampere?s law by computing? (a) the magnetic field due to a straight wire? (b) the magnetic field in the interior of a very long solenoid.? The contours
Show that a charged particle moving in the xy-plane in the presence of a magnetic field B = Bẑ will move in a circle. Compute the radius of the circle and frequency of rotation in terms of the
Use the magnetic force law and the definition of work? to show that magnetic forces do no work.? W = - dx · F la
Electrical power is often used to boil water for cooking.Here are the results of an experiment: a liter of water initially at 30 ◦C was boiled on an electric stove top burner. The burner is rated
An appliance that uses 1000W of power is connected by 12 gauge (diameter 2.053mm) copper wire to a120V (RMS) AC outlet. Estimate the power lost permeter, dPlost/dL, (in W/m) as resistive heating in
Consider two resistors placed in series, one after the other, in an electric circuit connected to a battery with voltage V. Show that the effective resistance of the pair is R1+R2 by using the fact
If each of the batteries used in the flashlight has an internal resistance of 0.5Ω (in series with the circuit), what fraction of power is lost to Joule heating within the batteries?
Consider an electric dipole composed of two charges ±Q at positions ± ξ/2. Write the exact electric field from the two charges and show that the leading term in an expansion in 1/r matches the
A cloud-to-ground lightning bolt can be modeled as a parallel plate capacitor discharge, with Earth’s surface and the bottom of the cloud forming the two plates. A particular bolt of lightning
The dielectrics in capacitors allow some leakage current to pass from one plate to the other. The leakage can be parameterized in terms of a leakage resistance RL. This limits the amount of time a
Suppose that a capacitor with capacitance C is charged to some voltage V and then allowed to discharge through a resistance R. Write an equation governing the rate at which energy in the capacitor
Starting from Gauss?s law and ignoring edge effects(i.e. assume that the plates are very large and the electric field is uniform and perpendicular to the plates),derive the formula for the
How much energy can you store on a parallel plate capacitor with d = 1 μm, A = 10 cm2, and ε = 100 ε0,assuming that the breakdown field of the dielectric is the same as for air?
Prove Gauss’s law from Coulomb’s law for static charge distributions by showing that the electric field of a single charge satisfies the integral form of Gauss’slaw and then invoking linearity.
The electric field outside a charged conducting sphere is the same as if the charge were centered at its origin.Use this fact to calculate the capacitance of a sphere of radius R, taking the second
In the American game of baseball, a pitcher throws a baseball, which is a round sphere of diameter b = 0.075 m, a distance of 18.4 m (60.5 feet), to a batter, who tries to hit the ball as far as he
Estimate Earth’s kinetic energy of rotation (the moment of inertia of a uniform sphere is 2/5 MR2).
Verify the assertion (see example 2.3) that Ekin =?1/2 V for the Moon in a circular orbit around Earth. [The magnitude of the centripetal acceleration for circular motion?can be rewritten a = v2/r.]
Estimate the rotational kinetic energy in a spinning yoyo (a plastic toy that you can assume is a cylinder of diameter 5.7 cm and mass 52 g, which rotates at 100 Hz). Compare to the gravitational
If the vehicle used as an example in this chapter accelerates to 50 km/h between each stop light, find the maximum distance between stoplights for which the energy used to accelerate the vehicle
One way to estimate the effective area of an object is to measure its limiting velocity v∞ falling in air. Explain how this works and find the expression for Aeff as a function of m (the
Consider an idealized cylinder of cross-sectional area A moving along its axis through an idealized diffuse gas of air molecules with vanishing initial velocity. Assume that the air molecules are
Compare the rate of power lost to air resistance for the following two vehicles at 60 km/h and 120 km/h: (a) General Motors EV1 with cdA ≅ 0.37m2, (b) Hummer H2 with cdA ≅ 2.45m2.

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