Model a hydrogen atom as an electron in a cubical box with side length L. Set the value of L so that the volume of the box equals the volume of a sphere of radius a = 5.29 x 10 -11 m,, the Bohr...
(a) Write out the ground-state electron configuration (1s 2 , 2s 2 ,) for the carbon atom. (b) What element of next-larger Z has chemical properties similar to those of carbon? Give the ground-state...
For magnesium, the first ionization potential is 7.6 eV. The second ionization potential (additional energy required to remove a second electron) is almost twice this, 15 eV, and the third ionization...
The energies for an electron in the K, L, and M shells of the tungsten atom are -69,500 eV, and -22,000 eV, respectively. Calculate the wavelengths of the K and K x rays of tungsten.
In terms of the ground-state energy, what is the energy E 1,1,1 , of the highest level occupied by an electron when 10 electrons are placed into a cubical box?
If the energy of the H 2 covalent bond is -4.48 eV, what wavelength of light is needed to break that molecule apart? In what part of the electromagnetic spectrum does this light lie?
(a) A molecule decreases its vibrational energy by 0.250 eV by giving up a photon of light. What wavelength of light does it give up during this process, and in what part of the electromagnetic...
When ultraviolet light with a wavelength of 400.0 nm falls on a certain metal surface, the maximum kinetic energy of the emitted photo electrons is measured to be 1.10 eV. What is the maximum kinetic...
The cathode-ray tubes that generated the picture in early color televisions were sources of x rays. If the acceleration voltage in a television tube is 15.0 kV, what are the shortest-wavelength x...
(a) What accelerating potential is needed to produce electrons of wavelength 5.00 nm? (b) What would be the energy of photons having the same wavelength as these electrons? (c) What would be the...
A certain atom has an energy level 3.50 eV above the ground state. When excited to this state, it remains 4.0 s, on the average, before emitting a photon and returning to the ground state. (a) What...
In a setup similar to that of Problem 35.41, the glass has an index of refraction of 1.53, the plates are each 8.00 cm long, and the metal foil is 0.015 mm thick. The space between the plates is...
When laser light of wavelength 632.8 nm passes through a diffraction grating, the first bright spots occur at 17.8 from the central maximum. (a) What is the line density (in lines cm) of this...
Although we have discussed single-slit diffraction only for a slit, a similar result holds when light bends around a straight, thin object, such as a strand of hair. In that case, a is the width of...
Two protons are moving away from each other. In the frame of each proton, the other proton has a speed of 0.600c. What does an observer in the rest frame of the earth measure for the speed of each...
A pencil that is 9.0 cm long is held perpendicular to the surface of a plane mirror with the tip of the pencil lead 12.0 cm from the mirror surface and the end of the eraser 21.0 cm from the mirror...
A person swimming 0.80 m below the surface of the water in a swimming pool looks at the diving board that is directly overhead and sees the image of the board that is formed by refraction at the...
The crystalline lens of the human eye is a double-convex lens made of material having an index of refraction of 1.44 (although this varies). Its focal length in air is about 8.0 mm, which also...
The cornea behaves as a thin lens of focal length approximately 1.8 cm, although this varies a bit. The material of which it is made has an index of refraction of 1.38, and its front surface is...
A person can see clearly up close but cannot focus on objects beyond 75.0 cm. She opts for contact lenses to correct her vision. (a) Is she nearsighted or farsighted? (b) What type of lens...
If the person in Problem 34.56 chooses ordinary glasses over contact lenses, what power lens (in diopters) does she need to correct her vision if the lenses are 2.0 cm in front of the eye? Problem...
Two identical audio speakers connected to the same amplifier produce in-phase sound waves with a single frequency that can be varied between 300 and 600 Hz. The speed of sound is 340 m/s. You find...
Two speakers that are 15.0 m apart produce in-phase sound waves of frequency 250.0 Hz in a room where the speed of sound is 340.0 m/s. A woman starts out at the midpoint between the two speakers. The...
If the entire apparatus of Exercise 35.11 (slits, screen, and space in between) is immersed in water, what then is the distance between the second and third dark lines? Exercise 35.11 Two slits...
If the eye receives an average intensity greater than 1.0 x 10 2 W/m 2 , damage to the retina can occur. This quantity is called the damage thresholdof the retina. (a) What is the largest average...
HeNe lasers are often used in physics demonstrations. They produce light of wavelength 633 nm and a power of 0.500 mW spread over a cylindrical beam 1.00 mm in diameter (although these quantities can...
A laser beam shines along the surface of a block of transparent material (see Fig. E33.8.). Half of the beam goes straight to a detector, while the other half travels through the block and then hits...
As shown in Fig. E33.11, a layer of water covers a slab of material Xin a beaker. A ray of light traveling upward follows the path indicated. Using the information on the figure, find (a) The index...
You want to support a sheet of fireproof paper horizontally, using only a vertical upward beam of light spread uniformly over the sheet. There is no other light on this paper. The sheet measures 22.0...
Light enters a solid pipe made of plastic having an index of refraction of 1.60. The light travels parallel to the upper part of the pipe (Fig. E33.17). You want to cut the face AB so that all the...
A piece of glass with a flat surface is at the bottom of a tank of water. If a ray of light traveling in the glass is incident on the interface with the water at an angle with respect to the normal...
Unpolarized light with intensity l 0 is incident on two polarizing filters. The axis of the first filter makes an angle of 60.0 with the vertical, and the axis of the second filter is horizontal....
(a) At what angle above the horizontal is the sun if sun-light reflected from the surface of a calm lake is completely polarized? (b) What is the plane of the electric-field vector in the reflected...
A ray of light traveling in a block of glass (n = 1.52) is incident on the top surface at an angle of 57.2 with respect to the normal in the glass. If a layer of oil is placed on the top surface of...
A 2.50-mH toroidal solenoid has an average radius of 6.00 cm and a cross-sectional area of 2.00 cm 2 . (a) How many coils does it have? (b) At what rate must the current through it change so that a...
A narrow beam of white light strikes one face of a slab of silicate flint glass. The light is traveling parallel to the two adjoining faces, as shown in Fig. E33.25. For the transmitted light inside...
Many biologically important molecules are optically active. When linearly polarized light traverses a solution of compounds containing these molecules, its plane of polarization is rotated. Some...
In a proton accelerator used in elementary particle physics experiments, the trajectories of protons are controlled by bending magnets that produce a magnetic field of 4.80 T. What is the...
(a) Take first and second derivatives with respect to time of q given in Eq. (30.28), and show that it is a solution of Eq (30.27). (b) At t = 0 the switch shown in Fig. 30.17 is thrown so that it...
A 84.0-nF capacitor is charged to 12.0 V, then disconnected from the power supply and connected in series with a coil that has L = 0.0420 H and negligible resistance. At an instant when the charge on...
You have a special light bulb with a very delicate wire filament. The wire will break if the current in it ever exceeds 1.50 A, even for an instant. What is the largest root-mean-square current you...
A capacitor is connected across an ac source that has volt-age amplitude 60.0 V and frequency 80.0 Hz. (a) What is the phase angle for the source voltage relative to the current? Does the source...
A short current element dt = (0.500 mm) j carries a current of 8.20 A in the same direction as dI. Point P is located at r = |(0.730 m) i + (0.390 m) k. Use unit vectors to express the magnetic field...
The body contains many small currents caused by the motion of ions in the organs and cells. Measurements of the magnetic field around the chest due to currents in the heart give values of about 10...
Certain bacteria (such as Aquaspirillum magnetotacticum) tend to swim toward the earths geographic north pole because they contain tiny particles, called magnetosomes, that are sensitive to a...
Currents in dc transmission lines can be 100 A or more. Some people have expressed concern that the electromagnetic fields (EMFs) from such lines near their homes could cause health dangers. For a...
A single loop of wire with an area of 0.0900 m 2 is in a uniform magnetic field that has an initial value of 3.80 T, is perpendicular to the plane of the loop, and is decreasing at a constant rate of...
A 10.0-cm-long solenoid of diameter 0.400 cm is wound uniformly with 800 turns. A second coil with 50 turns is wound around the solenoid at its center. What is the mutual inductance of the...
A straight, 2.5-m wire carries a typical household current of 1.5 A (in one direction) at a location where the earths magnetic field is 0.55 gauss from south to north. Find the magnitude and...
The largest cyclotron in the United States is the Tevatronat Fermilab, near Chicago, Illinois. It is called a Tevatron because it can accelerate particles to energies in the TeV range: 1 tera-eV = 10...
A parallel-plate air capacitor is to store charge of magnitude 240.0 pC on each plate when the potential difference between the plates is 42.0 V. (a) If the area of each plate is 6.80 cm 2 , what is...
One practical way to measure magnetic field strength uses a small, closely wound coil called a search coil. The coil is initially held with its plane perpendicular to a magnetic field. The coil is...
In a certain region of space, the magnetic field is not uniform. The magnetic field BÌ has both a z-component and a component that points radially away from or toward the z-axis. The...
Electric eels and electric fish generate large potential differences that are used to stun enemies and prey. These potentials are produced by cells that each can generate 0.10 V. We can plausibly...
You are working on an electronics project requiring a variety of capacitors, but you have only a large supply of 100-nF capacitors available. Show how you can connect these capacitors to produce each...
The thermal effciency of a natural gas boiler versus air/fuel ratio is plotted in Fig. E19.2. Using physical arguments, explain why a maximum occurs. Air-to-fuel stoichiometric ratio Figure E19.2...
An analyzer measures the pH of a process stream every 15 minutes. During normal process operation, the mean and standard deviation for the pH measurement are x = 5.75 and s = 0.05, respectively. When...
Consider the simple block diagram for the feedback control of circadian rhythms (i.e., 24 h) in Fig. 24.5.To simplify things considerably, assume that transcription obeys a first order process with...
Water (F 1 ) is blended with a stream F 2 with 40% ethanol to make a whiskey product that is 30% ethanol. Assume F 1 = 4 gal/min and F 2 = 4 gal/min. (a) Develop a steady-state material balance model...
In a computer control system, the high and low warning limits for a critical temperature measurement are set at the 2-sigma limits, T 3 T , where T is the nominal temperature and T is the estimated...
In order to improve the reliability of a critical control loop, it is proposed that redundant sensors be used. Suppose that three independent sensors are employed and each sensor works properly 95%...
The vector from the origin to point A is given as (6,2,4), and the unit vector directed from the origin toward point B is (2,2, 1)/3. If points A and B are ten units apart, find the coordinates of...
A circle, centered at the origin with a radius of 2 units, lies in the xy plane. Determine the unit vector in rectangular components that lies in the xy plane, is tangent to the circle at ( 3,1, 0),...
A vector field is specified as G = 24xya x + 12(x 2 + 2)a y + 18z 2 a z . Given two points, P(1, 2,1) and Q(2, 1, 3), find (a) G at P; (b) A unit vector in the direction of G at Q; (c) A unit vector...
For the transfer functions (a) Derive an analytical expression for the step response to a unit step change. Evaluate the step-response coefficients, {S i }, for a sampling period of Ît = 1. (b)...
A process (including sensor and control valve) can be modeled by the transfer function, (a) Derive an analytical expression for the response to a unit step change in the input. (b) Suppose that the...
Consider the fermentor problem in Example 23.1. (a) Design an IMC controller for the first operating point (dilution = 0.202 h -1 ), and simulate the response to both a +0.5 [g/L] and a -0.5 [g/L]...
Consider the granulation model that was given in Example 23.2. (a) Design an MPC controller, using the nominal process model. Initially consider a control horizon of M = 2 and a prediction horizon of...
Gaohua and Kimura (2006) derived an empirical patient model for the manipulation of ambient temperature u ( C) to inï¬uence the patients brain intracranial pressure (ICP) y (mm Hg). The...
In a rehabilitation training experiment for a neurological patient, a step change in treadmill speed of +2.5 km/h was made. The patient heart rate response HR is given in Fig. E23.4. (a) Derive an...
A crystallize is used to separate a pharmaceutical product from the fermentation extract. The three manipulated variables are the fines dissolution rate u 1 , the crystallize temperature u 2 , and...
Given the points M(0.1,0.2,0.1), N(0.2, 0.1, 0.3), and P(0.4, 0, 0.1), find (a) The vector R MN ; (b) The dot product R MN R MP ; (c) The scalar projection of R MN on R MP ; (d) The angle between R...
Write an expression in rectangular components for the vector that extends from (x 1 , y 1 , z 1 ) to (x 2 , y 2 , z 2 ) and determine the magnitude of this vector.
Find (a) The vector component of F = 10a x 6a y + 5a z that is parallel to G = 0.1a x + 0.2a y + 0.3a z ; (b) The vector component of F that is perpendicular to G; (c) The vector component of G that...
Consider the diabetic patient in Example 23.3. Your goal is to design an automated device to administer insulin infusion in response to meal disturbances. (a) Considering only the insulin-glucose...
Recall the artificial pancreas control problem from Section 23.2. In order to regulate blood glucose BG, you have two possible manipulated variables to choose from: pumping insulin F I and pumping...
Data from an actual clinical trial for the closed-loop artificial pancreas is summarized in the following cumulative histograms, where the percentage of blood glucose measurements below a particular...
Express the unit vector ax in spherical components at the point: (a) r = 2, = 1 rad, = 0.8 rad; (b) x = 3, y = 2, z = 1; (c) = 2.5, = 0.7 rad, z = 1.5.
Let a point charge Q 1 = 25 nC be located at P 1 (4,2, 7) and a charge Q 2 = 60 nC be at P 2 (3, 4,2). (a) If = 0, find E at P 3 (1, 2, 3). (b) At what point on the y axis is E x = 0?
In this exercise, treat the components as simple (reactive) chemical species and perform the appropriate (dynamic) material balance. Assume that a messenger RNA mRNA is produced by a constant (basal)...
Consider the simplified version of the chemotaxis circuit in Fig. 24.10. (a) Derive the conditions for the process gain K that ensure that the receptor activity is always reset to zero and even for...
As a specific biological example for Exercise 24.2 and Fig. E24.2(b), the synthesis of tryptophan can be described by the following set of material balances: where k 1 , k 2 , k 3 , and k 4 represent...
An interesting motif in biological circuits is a switch, in which the system can change from (effectively) one binary state to another. An analysis of a continuous reaction network reveals a rise to...
Recall the Central Dogma for biology, which is depicted in simplified form in the block diagram Fig. E24.7. In this case, it is assumed that nuclear transport can be ignored. Let X denote an...
A potential field in free space is expressed as V = 20/(xyz)V. (a) Find the total energy stored within the cube 1 < x, y, z < 2. (b) What value would be obtained by assuming a uniform energy density...
Let J = 25/a 20/( 2 + 0.01) a z A/m 2 . (a) Find the total current crossing the plane z = 0.2 in the a z direction for < 0.4. (b) Calculate /t. (c) Find the outward current crossing the closed...
Let J = 400 sin /(r 2 + 4) a r A/m 2 . (a) Find the total current flowing through that portion of the spherical surface r = 0.8, bounded by 0.1 < < 0.3, 0 < < 2. (b) Find the average value of J...
Region 1 (x 0) is a dielectric with r1 = 2, while region 2(x < 0) has r2 = 5. Let E 1 = 20a x 10a y + 50a z V/m. (a) Find D 2 . (b) Find the energy density in both regions.
Two perfect dielectrics have relative permittivities r1 = 2 and r2 = 8. The planar interface between them is the surface x y + 2z = 5. The origin lies in region 1. If E 1 = 100a x + 200a y 50a z...
Electron and hole concentrations increase with temperature. For pure silicon, suitable expressions are h = e = 6200T 1.5 e 7000 /T C/m 3 . The functional dependence of the mobilities on temperature...
A dipole with p = 0.1a z C m is located at A(1, 0, 0) in free space, and the x = 0 plane is perfectly conducting. (a) Find V at P(2, 0, 1). (b) Find the equation of the 200 V equipotential surface...
Let the surface y = 0 be a perfect conductor in free space. Two uniform infinite line charges of 30 nC/m each are located at x = 0, y = 1, and x = 0, y = 2. (a) Let V = 0 at the plane y = 0, and find...
Let V = 10( + 1)z 2 cos V in free space. (a) Let the equipotential surface V = 20 V define a conductor surface. Find the equation of the conductor surface. (b) Find and E at that point on the...
Solve Problem B.1 by GaussSeidel iteration. Data from problem B.1 Determine the solution of the following simultaneous equations by the inverse method. 1x 1 + 3x 2 = 5 4x 1 - 1x 2 = 12
A continuous data signal is quantized and transmitted using a PCM system. If each data sample at the receiving end of the system must be known to within 0.25% of the peak-to-peak full-scale value,...
Write a computer program to find the bandwidth of a lowpass energy signal that contains a certain specified percentage of its total energy, for example, 95%. In other words, write a program to find
Solve the following system of simultaneous equations by Gaussian elimination. x 1 + 4x 2 - 5x 3 = 4 3x 2 + 4x3 = -1 -2x 1 -1x 2 + 2x 3 = -3
Determine the solution of the following simultaneous equations by the inverse method. 1x 1 + 3x 2 = 5 4x 1 - 1x 2 = 12
Given that x 1 = 2y 1 - y 2 z 1 = -x 1 - x 2 x 2 = y 1 - y 2 z 2 = 2x 1 + x 2 a. Write these relationships in matrix form. b. Express [z] in terms of {y}. c. Express [y] in terms of [z].