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All Textbook Solutions for University Physics Volume 2

Show that the self-inductance per unit length of an infinite, straight, thin wire is infinite.Two long, parallel wires cy equal currents in opposite directions. The radius of each wire is a, and the distance between the centers of the wires is d, Show that if the magnetic flux within the wires themselves can be ignored, the self-inductance of a length 1 of such a pair of wires is L 0 l in d1 a (Hint Calculate the magnetic flux through a rectangle of length 1 between the wires and then use L=N/I A small, rectangular single loop of wire with dimensions 1, and a is placed, as shown in the plan of a much larger, rectangular single loop of wire. The two short sides of the larger loop axe so far from the smaller loop that their magnetic fields over the smaller fields over the smaller loop can be ignored. What is the mutual inductance of the two loops?Suppose that a cylindrical solenoid is wrapped around a core of iron whose magnetic susceptibility is x. Using Equation 14.9, show that the self-inductance of the solenoid is given by L=(l+x)+0n2Al where l is its length, A its cross-sectional area and N its total number of turns. Nm=U A solenoid with 4 x 107turns/m has an iron core placed in it whose magnetic susceptibility is 4.0 x 103. (a) If a cent of 2.0 A flows through the solenoid, what is the magnetic field in the iron core? (b) What is the effective surface current formed by the aligned atomic current loops in the iron core? (c) What is the self-inductance of the filled solenoid?A rectangular toroid with inner radius R1= 7.0cm, outer radius R2= 9.0cm, height h = 3.0, and N=3.0, and N = 3000 turns is filled with an iron core a magnetic susceptibility 5.2 × 103. (a) What is the self-inductance of the toroid? (b) If the current through the toroid is 2.0 A, what is the magnetic field at the center of the core? (c) For this same 2.0-A current, what is the effective surface current formed by the aligned atomic current loops in the iron core?The switch S of the circuit shown below is closed t =0. Determine (a) the initial current through the battery and stead-state current the battery.In an oscillating RLC circuit, R = 7.0 L. = 10 mH. And C = 3.0 F. Initially, the capacitor has a charge of 8.0 C and the current is zero. Calculate the charge on the capacitor (a) five cycles later and (b) 50 cycles later.A 25.0-H inductor has 100 A of current turned off in 1.00 ms . (a) What voltage is induced to oppose this? (b) What is unreasonable about this result? (c) Which assumption or premise is responsible’A coaxial cable has an inner conductor of radius a, and outer thin cylindrical shell of radius b. A current I flows in the inner conductor and returns in the outer conductor. The self-inductance of the structure will depend on bow the current in the inner cylinder tends to be distributed. Investigate the following two extreme cases. (a) Let current in the inner conductor be distributed only on the surface and find the self-inductance. (b) Let current in the inner cylinder be distributed uniformly over its cross-section and find the self-inductance. Compare with your results in (a).In a damped oscillating circuit the energy is dissipated In the resistor. The Q-factor Is a measure of the persistence of the oscillator against the dissipative loss. (a) Prove that for a ligt1y danced circuit the energy, U, in the circuit decreases according to the following equation. du dt =2u,where= R 2L (b) Using the definition of the Q-factor as energy divided by the loss over the next cycle, prove that Q-factor of a lightly damped oscillator as defined in this problem is Q= U begin U onecycle = 1 R L C c (Hint: For (b), to obtain Q, divide E at the beginning of one cycle by the change E over the next cycle.)The switch in the circuit shown below is closed at t = 0s. Find currents through (a) R1(b) R2, and (C) the battery as function of time.A square loop of side 2 cm is placed 1 cm from a long wire carrying a current that varies with time at a constant rate of 3 A/s as shown below. (a) Use Ampere’s law and find the magnetic field as a function of time from the current in the wire. (b) Determine the magnetic flux through the loop. (c) If the loop has a resistance of 3 how much induced current flows in the loop?A rectangular copper ring, of mass 100 g and resistance 0.2 1, is in a region of uniform magnetic field that is perpendicular to the area enclosed by the ring and horizontal to Earth’s surface. The ring is let go from rest when it is at the edge of the nonzero magnetic field region (see below). (a) Find its speed when the ring just exits the region of uniform magnetic field. (b) If it was let go at t = 0, what is the time when it exits the region of magnetic field for the following values: a=25cm,b=50cmB=3T , and g=9.8m/s2 ? Assume the magnetic field of the induced current is negligible compared to 3T.Check Your Understanding If a European ac voltage source is considered, what is the time difference between the zero crossings on an ac voltage-versus-time graph?Check Your Understanding Repeat Example 15.1 for an ac source of amplitude 20 V and frequency 100 Hz.Check Your Understanding Find the voltages across the resistor, the capacitor, and the inductor in the circuit of Figure 15.11using v(t)=v0sint asthe output of the ac generator.Check Your Understanding An ac voltmeter attached across the terminals of a 45-Hz ac generator reads 7.07 V Write an expression for the emf of the generator.Check Your Understanding Show that the tins voltages across a resistor, a capacitor, and an inductor in an ac circuit where the tins current is Irmsare given by IrmsR,IrmsXC , and IrmsXL, respectively. Determine these values for the components of the RLC circuit of Equation 15.12. Pave=12t0v0cos (15.12)Check Your Understanding In the circuit of Figure 15.11, L=2.0103H,C=5.0104F , and R=40 (a) What is the resonant frequency? (b) What Is the Impedance of the circuit at resonance? (c) If the voltage amplitude is 10 V, what is i(t) at resonance? (d) The frequency of the AC generator is now changed to 200 Hz. Calculate the phase difference between the current and the emf of the generator. The ac circuit shown in Figure 15.11, called an RLC series circuit, is a series combination of a resistor, capacitor, and inductor connected across an ac source. It produces an emf of Figure 15.11 (a) An RLC series circuit, (b) A comparison of the generator output voltage and the current the value of the phase difference depends on the values of R, C, and L.Check Your Understanding What happens to the resonant frequency of an RLC series circuit when the following quantities are increased by a factor of 4: (a) the capacitance, (b) the self-inductance, and (c) the resistance?Check Your Understanding The resonant angular frequency of an RLC series circuit is 4.0102 rad/s. An ac source operating at this frequency transfers an average power of 2.0102W to the circuit. The resistance of the circuit Is 0.50 Q. Write an expression for the emf of the source.Check Your Understanding A transformer steps the line voltage down from 110 to 9.0 V so that a current of 0.50 A can be delivered to a doorbell (a) What is the ratio of the number of turns in the primary and secondary windings? (b) What is the current in the primary winding? (c) What is the resistance seen by the 110-V source?What is the relationship between frequency and angular frequency?Explain why at high frequencies a capacitor acts as an ac short, whereas an inductor acts as an open circuit.In an RLC series circuit, can the voltage measured across the capacitor be greater than the voltage of the source? Answer the same question for the voltage across the inductor.For what value of the phase angle between the voltage output of an ac source and the current is the average power output of the source a maximum?Discuss the differences between average power and instantaneous power.The average ac current delivered to a circuit is zero. Despite this, power is dissipated in the circuit. Explain.Can the instantaneous power output of an ac source ever be negative? Can the average power output be negative?The power rating of a resistor used in ac circuits refers to the maximum average power dissipated in the resistor. How does this compare with the maximum instantaneous power dissipated in the resistor?Why do transmission lines operate at very hig1 voltages while household circuits operate at fairly small voltages?How can you distinguish the primary winding from the secondary winding in a step-up transformer?Battery packs in some electronic devices are charged using an adapter connected to a wall socket. Speculate as to the purpose of the adapter.Will a transformer work if the input is a dc voltage?Why are the primary and secondary coils of a transformer wrapped around the same closed loop of iron?Write an expression for the output voltage of an ac source that has an amplitude of 12 V and a frequency of 200 Hz.Calculate the reactance of a 5.0F capacitor at (a) 60 Hz, (b) 600 Hz, and (c) 6000 Hz.What is the capacitance of a capacitor whose reactance is 10 at 60 Hz?Calculate the reactance of a 5.0-mH inductor at (a) 60 Hz, (b) 600 Hz, and (C) 6000 Hz.What is the self-inductance of a coil whose reactance is 10 at 60 Hz?At s1iat frequency is the reactance of a 20F capacitor equal to that of a 10-mH inductor?At 1000 Hz, the reactance of a 5.0-mH inductor is equal to the reactance of a particular capacitor. What is the capacitance of the capacitor?A 50resistor is connected across the emf v(t)=(160V)sin(120t) . Write an expression for the current throug1 the resistor.A 25F capacitor is connected to an emf given by v(t)=(160V)sin(120t) . (a) What is the reactance of the capacitor? (b) Write an expression for the current output of the source.A 100-mH inductor is connected across the emf of the preceding problem. (a) What is the reactance of the Inductor? (b) Write an expression for the current through the inductor.What is the impedance of a series combination of a 50resistor, a 5.0F capacitor, and a 10F capacitor at a frequency of 2.0 kHz?A resistor and capacitor are connected in series across an ac generator. The emf of the generator is given by V(t)=V0cost , where V0=120V , =120rad/s , R=400and C=400F.(a)What is the impedance of the circuit? (b) What is the amplitude of the current through the resistor? (C) Write an expression for the current through the resistor. (d) Write expressions representing the voltages across the resistor and across the capacitor.A resistor and inductor are connected in series across an ac generator. The emf of the generator is given by v(t)=V0cost , where V0=120V and =120rad/s ; also, R=400 and L = 1.5 H. (a) What Is the impedance of the circuit? (b) What is the amplitude of the current through the resistor? (C) Write an expression for the current through the resistor. (d) Write expressions representing the voltages across the resistor and across the inductor.In an RLC series circuit, the voltage amplitude and frequency of the source are 100 V and 500 Hz, respectively, an R = 5O0. L=0.20H, and C=2.0F . (a)What is the impedance of the circuit? (b) What is the amplitude of the current from the source? (C) If the emf of the source Is given by v(tt)=(100V)sin , how does the current vary with time? (d) Repeat the calculations with C changed to 0.20F .An PLC series circuit with R=600 , L = 30 mH. and c=0.050F is driven by an ac source whose frequency and voltage amplitude are 500 Hz and 50 V, respectively, (a) What is the impedance of the circuit? (b) What is the amplitude of the current in the circuit? (c) What is the phase angle between the emf of the source and the current?For the cucu3t shown below, what are (a) the total impedance and (b) the phase angle between the current and the emf? (C) Write, an expression for i(t).The emf of an ac source is given by v(t)=v0sint. where V0=100V and at =200rad/s . Calculate the average power output of the source if it is connected across (a) a 20F capacitor, (b) a 20-mH inductor, and (c) a 50 resistor.Calculate the rms currents for an ac source is given by v(t)=v0sint , where V0=100V and =200rad/s when connected across (a) a 20F capacitor, (b) a 20-mH inductor, and (c) a 50 resistor.A 40-mH inductor is connected to a 60-Hz AC source whose voltage amplitude is 50 V. If an AC voltmeter is placed across the inductor, what does it read?For an RLC series circuit, the voltage amplitude and frequency of the source are 100 V and 500 Hz, respectively; R=500 ; and L = 0.20H . Find the average power dissipated in the resistor for the following values for the capacitance: (a) C=2.0F and (b) C=2.0F .An ac source of voltage amplitude 10 V delivers electric energy at a rate of 0.80 W when its current output is 2.5 A. What is the phase angle between the emf and the current?An RLC series circuit has an impedance of 60 and a power factor of 0.50, with the voltage Lagging the current (a) Should a capacitor or an inductor be placed in series with the elements to raise the power factor of the circuit? (b) What is the value of the capacitance or self-inductance that will raise the power factor to unity?Calculate the resonant angular freqency of an RLC series circuit for which R=20 , L=75mH, and C=4.0F , (b) If R is changed to 300, what happens to the resonant angular frequency?The resonant frequency of RLC series circuits is 2.0103Hz , If the self-inductance in the circuit is 5.0 mH, what is the capacitance in the circuit?(a) What is the resonant frequency of an RLC series circuit with R=20 , L=2.0mH , and C=4.0F (b) What is the impedance of the circuit at resonance ?For an RLC series circuit, R=100 , L = 150 mH, and C=0.25F . (a) If an ac source of variable frequency is connected to the circuit, at what frequency is maximum power dissipated in the resistor? (b) What is the quality factor of the circuit?An ac source of voltage amplitude 100 V and variable frequency f drives an RLC series circuit with R=10 , L = 2.0 mH, and C=25F (a) Plot the canal through the resistor as a function of the frequency f. (b) Use the plot to determine the resonant frequency of the circuit.(a) What is the resonant frequency of a resistor, capacitor, and inductor connected in series if R = 100 . L = 2.0 H, and C = 5.0 F ? (b) If this combination is connected to a 100-V source operating at the resonant frequency, what is the power output of die source? (c) What is the Q of the circuit? (d) What is die bandwidth of the circuit?Suppose a coil has a self-inductance of 20.0 H and a resistance of 200 . What (a) capacitance and (b) resistance must be connected in series with the coil to produce a circuit that has a resonant frequency of 100 Hz and a Q of 10?An ac generator is connected to a device whose internal circuits are not known. We only know current and voltage outside the device, as shown below. Based on the information given, whit can you infer about the electrical nature of the device and its power usage?A step-up transformer is designed so that the output of its secondary winding is 2000 V(rms) when the primary winding is connected to a 110-V (rms) line voltage, (a) If there are 100 turns in the primary winding, how many turns are there in the secondary winding? (b) If a resistor connected across the secondary winding draws an rms current of 0.75 A. what is the current in die primary winding?A step-up transformer connected to a 100-V line U used to supply a hydrogen-gas discharge tube with 5.0 kV (rms). The tube dissipates 75 W of power, (a) What is the ratio of the number of turns in the secondary winding to the number of turns in the primary winding? (b) What are the nns currents in the primary and secondary windings? (c) What is the effective resistance seen by the 110-V source?An ac source of emf delivers 5.0 mW of power at an rms current of 2.0 mA when it is connected to the primary cod of a transformer. The rms voltage across the secondary coil is 20 V. (a) What are die voltage across the primary coil and the current through the secondary coil? (b) What is the ratio of secondary to primary turns for the transformer?A transformer is used to down 110 V from a wall socket to 9.0 V for a ro. (a) If the primary winding has 500 turns, how many s does the secondary winding have? (b) If the radio operates at a current of 500 mA, what is the current 500 mA, what is the current through the primary winding?A transformer is used to supply a 12-V model train with power from a 110-V wall plug. The train operates at 50 W of power, (a) What is the rms current in the secondary coil of the transformer? (b) What is me rms current in the primary coil? (c) What is the ratio of the number of primary to secondary turns? (d) What is the resistance of the tram? (e) What is the resistance seen by the 110-V source?The emf of an ac source is given by v(t)=V0sint, where V0=100V and =200 . Find an expression that represents the output current of the source if it is connected across (a) a 20-pF capacitor, (b) a 20-mH inductor, and (c) a 50 resistor.A 700-pF capacitor is connected across an ac source with a voltage amplitude of 160 V and a frequency of 20 kHz. (a) Determine the capacitive reactance of the capacitor and the amplitude of the output current of the source. (b) If the frequency is changed to 60 Hz while keeping the voltage amplitude at 160 V, what are the capacitive reactance and the current amplitude?A 20-mH inductor is connected across an AC source with a variable frequency and a constant-voltage amplitude of 9.0 V. (a) Determine the reactance of the circuit and the maximum current through the inductor when the frequency Is set at 20 kHz. (b) Do the same calculations for a frequency of 60 Hz.A 30F capacitor is connected across a 60-Hz ac source whose voltage amplitude is 50 V. (a) What is the maximum charge on the capacitor? (b) What is the maximum current into the capacitor? (c) What is the phase relationship between the capacitor charge and the current in the circuit?A 7.0-mH induct is connected across a 60-Hz ac source whose voltage amplitude is 50 V. (a) What is the maximum current through the inductor? (b) What is the phase relationship between the current through and the potential difference across the inductor?What is the impedance of an RLC series circuit at the resonant frequency?What is the resistance R in the circuit shown below if the amplitude of the ac through the inductor is 4.24 A?An ac source of voltage amplitude 100 V and frequency 1.0 kHz drives an PLC series circuit with R=20, L = 4.0 mH, and C=50F . (a) Determine the rms current through the circuit, (b) What are the rms voltages across the three elements? (c) What is the phase angle between the emf and the current? (d) What is the power output of the source? (e) What is the power dissipated in the resistor?In an RLC series circuit, R=200,L=1.0H , C=50F,v0=120V , and f = 50 Hz. What is the power output of the source?A power plant generator produces 100 A at 15 kV (rms). A transformer is used to step up the transmission line voltage to 150 kV (rms). (a) What is rms current in the transmission line? (b) If the resistance per unit length of the line is 8.6108/m , what is the power loss per meter in the line? (c) What would the power loss per meter be If the line voltage were 15 kV (rms)?Consider a power plant located 25 km outside a town delivering 50 MW of power to the town. The transmission lines are made of aluminum cables with a 7cm2 cross- sectional area. Find the loss of power In the transmission lines if it is transmitted at (a) 200 kV (rms) and (b) 120 V (rms).Neon signs require 12-kV for their operation. A transformer is to be used to change the voltage from 220-V (rms) ac to 12-kV (rms) ac. What must the ratio be of turns in the secondary winding to the turns in the primary winding? (b) What is the maximum rms current the neon lamps can draw if the fuse in the primary winding goes off at 0.5 A? (C) How much power is used by the neon sign when it is drawing the maximum current allowed by the fuse in the primary winding?The 335-kV ac electricity from a power transmission line is fed into the primary winding of a transformer. The ratio of the number of turns in the secondary winding to the number in the primary winding is Ns/Np=100 . (a) What voltage is induced in the secondary winding? (b) What is unreasonable about this result? (c) Which assumption or premise is responsible?A 1.5k resistor and 30-mH inductor are connected in series, as below, across a120-V(rms)ac power source oscillating at 60-Hz frequency. (a) Find the current in the circuit. (b) Find the voltage drops across the resistor and inductor. (C) Find the impedance of the circuit. (d) Find the power dissipated in the resistor. (e) Find the power dissipated in the inductor. (1) Find the power produced by the source.A 20 resistor, 50F capacitor, and 30-mH inductor are connected in series with an ac source of amplitude 10 V and frequency 125 Hz. (a) What is the impedance of the circuit? (b) What is the amplitude of the current in the circuit? (c) What is the phase constant of the current? Is it leading or lagging the source voltage? (d) Write voltage drops across the resistor, capacitor, and inductor and the source voltage as a function of time, (e) What is the power factor of the circuit? (f) How much energy is used by the resistor in 2.5 s?A 200- resistor, 150- F capacitor, and 23-H inductor are connected in series with an ac source of amplitude 10 V and variable angular frequency . (a) What is the value of the resonance frequency R? (b) What is the amplitude of the current if =R ? (c) What is the phase constant of the current when =R ? Is it leading or lagging the source voltage, or is It In phase? (d) Write an equation for the voltage drop across the resistor as a function of time when =R . (e) What is the power factor of the circuit when =R ? (f) How much energy is used up by the resistor in 2.5s when =R ?Find the reactances of the following capacitors and inductors in ac circuits with the given frequencies in each case: (a) 2-mH inductor with a frequency 60-Hz of the ac circuit; (b) 2-mM Inductor with a frequency 600-Hz of the ac circuit; (C) 20-mM inductor with a frequency 6-Hz of the ac circuit; (d) 20-mM inductor with a frequency 60-Hz of the ac circuit; (e) 2-mF capacitor with a frequency 60-Hz of the ac circuit; and (f) 2-mF capacitor with a frequency 600-Hz of the AC circuit.An output impedance of an audio amplifier has an impedance of 500 and has a mismatch with a low-impedance 8- loudspeaker. You are asked to insert an appropriate transformer to match the impedances. What turns ratio will you use, and why? Use the simplified circuit shown below.Show that the SI unit for capacitive reactance is the ohm. Show that the SI unit for inductive reactance is also the ohm.A coil with a self-inductance of 16 mH and a resistance of 6.0 is connected to an ac source whose frequency can be varied. At what frequency will the voltage across the coil lead the current through the coil by 45 ?An RLC series circuit consists of a 50 resistor, a 200F capacitor, and a 120-mN inductor whose coil has a resistance of 20. The source for the circuit has an tins emf of 240 V at a frequency of 60 Hz. Calculate the tins voltages across the (a) resistor, (b) capacitor, and (c) inductor.An RLC series circuit consists of a 10 resistor, an 8.0F capacitor, and a 50-mH inductor. A 110-V (rms) source of variable frequency is connected across the combination. What is the power output of the source when its frequency is set to one-half the resonant frequency of the circuit?Shown below are two circuits that act as crude high-pass filters. The input voltage to the circuits is vin, and the output voltage is vout. (a) Show that for the capacitor circuit, voutvin=11+1/2R2C2 and for the inductor circuit, voutvin=LR2+2L2 (b) Show that for high frequencies, voutvin , but for low frequencies, vout0 .The two circuits shown below act as crude low-pass filters. The input voltage to the circuits is vin, and the output voltage is vout. (a) Show that for the capacitor circuit, voutvin=11+2R2C2 and for the inductor circuit, voutvin=RR2+2L2 (b) Slow that for low frequencies, voutvin , but for high frequencies, vout0 .Check Your Understanding When the emf across a capacitor Is turned on and the capacitor Is allowed to charge, when does the magnetic field Induced by the displacement current have the greatest magnitude?Check Your Understanding Could a purely electric field propagate as a wave through a vacuum without a magnetic field? Justify your answer.Check Your Understanding The wave equation was obtained by (1) finding the E field produced by the changing B field. (2) finding the B field produced by the changing E field, and combining the two results. Which of Maxwell’s equations was the basis of step (1) and which of step (2)?Check Your Understanding What conclusions did our analysis of Maxwell's equations lead to about these properties of a plane electromagnetic wave: (a) the relative directions of wave propagation, of the E field, and of B field, (b) the speed of travel of the wave and how the speed depends on frequency, and (c) the relative magnitudes of the E and B fields.Check Your Understanding How would the speed and acceleration of a radiation-propelled spacecraft be affected as it moved farther from the Sun on an interplanetary space flight?Check Your Understanding How do the electromagnetic waves f the different kinds of electromagnetic radiation differ?Explain how the displacement current maintains the continuity of current in a circuit containing a capacitor.Describe the field lines of the induced magnetic field along the edge of the imaginary horizontal cylinder shown below if the cylinder is in a spatially uniform electric field that is horizontal, pointing to the right, and increasing in magnitude.Why is it much easier to demonstrate in a student lab that a changing magnetic field induces an electric field than it is to demonstrate that a changing electric field produces a magnetic field?If the electric field of an electromagnetic wave is oscillating along the z-axis and the magnetic field is oscillating along the x-axis, in what possible direction is the wave traveling?In which situation shown below will the electromagnetic wave be more successful in inducing a current in the wire? Explain.In which situation shown below will the electromagnetic wave be more successful in inducing a current in the loop? Explain.Under what conditions might wires in a circuit wher the current flows in only one direction emit electromagneti waves?Shown below is the interference pattern of two radio antennas broabcasting the same signal. Explain how this is analogous to the interference pattern for sound produced by two speakers. Could this be used to make a directional antenna system that broadcasts preferentially in certain directions? Explain.When you stand outdoors in the sunlight, y can you feel the energy that the sunlight carries, but not the momentum it carries?How does the intensity of an electromagnetic wave depend on its electric field? How does it depend on its magnetic field?What is the physical significance of the Poynting vector?A 2.0-mW helium-neon laser transmits a continuous beam of red light of cross-sectional area 0.25 cm2. If the beam does not diverge appreciably, how would its rms electric field vary with distance from the laser? Explain.Why is t1 radiation pressure of an electromagnetic wave on a perfectly reflecting surface twice as large as the pressure on a perfectly absorbing surface?Why did the early Hubble Telescope photos of Cornet Ison approaching Earth show it to have merely a fuzzy coma around it, and not the pronounced double tail that developed later (see below)? Figure 16.21. (credit of work by NSA, ESA J.-Y Li (Planetary Science Institute), and the Hubble Comet ISON Imaging Science Team)(a) If the electric field and magnetic field in a sinusoidal plane wave were interchanged, in Which direction relative to before would the energy propagate? (b) What if the electric and the magnetic fields re both changed to their negatives?Compare the speed, wavelength, and frequency of radio waves and X-rays traveling in a vacuum.Accelerating electric charge emits electromagnetic radiation. How does this apply in each case: (a) radio waves, (b) infrared radiation.Compare and contrast the meaning of the prefix “micro” in the names of SI units in the term microwaves.Part of the light passing through the air is scattered in all directions by the molecules comprising the atmosphere. The wavelengths of visible Light are larger than molecular sizes, and the scattering is strongest for wavelengths of light closest to sizes of molecules. (a) Which of the main colors of light is scattered the most? (b) Explain why this would give the sky its familiar background color at midday.When a bowl of soup is removed from a microwave oven, the soup is found to be steaming hot, s1wreas the bowl is only warm to the touch. Discuss the temperature changes that have occurred in terms of energy transfer.Certain orientations of a broadcast television antenna give better reception than others for a particular station. Explain.What property of light corresponds to loudness in sound?Is the visible region a major portion of the electromagnetic spectrum?Can the human body detect electromagnetic radiation that is outside the visible region of the spectrum?Radio waves normally have their E and B fields in specific directions, whereas visible light usually has its E and B fields in random and rapidly changing directions that are perpendicular to each other and to the propagation direction. Can you explain why?Give an example of resonance in the reception of electromagnetic waves.Illustrate that the size of details of an object that can be detected with electromagnetic waves is related to their wavelength, by comparing details observable with two different types (for example, radar and visible light).In which pan of the electromagnetic spectrum are each of these waves: (a) f=10.0kHz , (b) f==750nm , (C) f=1.25108Hz , (d) 0.30 nm In what range of electromagnetic radiation are the electromagnetic waves emitted by power lines in a country that uses 50-Hz ac current?If a microwave oven could be modified to merely tune the waves generated to be in the infrared range instead of using microwaves, how would this affect the uneven heating of the oven?A leaky microwave oven in a home can sometimes cause interference with the homeowner’s WIFI system. Why?When a television news anchor in a studio speaks to a reporter in a distant country, there is sometimes a noticeable lag between when the anchor speaks in the studio and when the remote reporter hears it and replies. Explain what causes this delay.Show that the magnetic field at a distance r from the axis of two circular parallel plates, produced by placing charge Q(t) on the plates is Bind=02rdQ(t)dt Express the displacement current in a capacitor in terms of the capacitance and the rate of change of the voltage across the capacitor.A potential difference V(t) = V0sin tis maintained across a parallel-plate capacitor with capacitance C consisting of two circular parallel plates. A thin wire with resistance R connects the centers of the two plates, allowing charge to leak between plates while they are charging. (a) Obtain expressions for the leakage current Ires(t) in the thin wire. Use these results to obtain an expression for the current Ireal(t) in the wires connected to the capacitor. (b) Find the displacement current in the space between the plates from the changing electric field between the plates. (c) Compare Ireal(t) with the sum of the displacement current Id(t)and resistor current Ires(t) between the plates, and explain why the relationship you observe would be expected.Suppose the parallel-plate capacitor shown below is accumulating charge at a rate of 0.010 C’s. What is the induced magnetic field at a distance of 10 cm from the capacitator?The potential difference V(t) between parallel plates shown above is instantaneously increasing at a rate of 107V/s. What is the displacement current between the plates if the separation of the plates is 1.00 cm and they have an area of 0.200m2?A parallel-plate capacitor has a plate area of A=0.250m2 and a separation of 0.0100 in. What must be must be the angular frequency w for a voltage V(t)=V0sint with V0=100V to produce a maximum displacement displacement current of 1.00 A between the plates?The voltage across a parallel-plate capacitor with area A=800cm2 and separation d = 2 mm varies sinusoidally as V = (15 mV) cos (150t), where t is in seconds. Find the displacement current between the plates.The voltage across a parallel-plate capacitor with area A and separation d varies with time t as V = at2, where a is a constant. Find the displacement current between the plates.If the Sun suddenly turned off, we would not know it until its light stopped coming. How long would that be, given that the Sun is 1.4961011 away?What is the maximum electric field strength in an electromagnetic wave that has a maximum magnetic field strength of 5.00104 (about 10 times Earth's magnetic field)?An electromagnetic wave has a frequency of 12 MHz. What is its wavelength in vacuum?If electric and magnetic field strengths vary sinusoidally in time at frequency 1.00 GHz, being zero at t=0, then E=E0sin2ft. (a) When are the field strengths next equal to zero? (b) 4Jhen do they reach their most negative value? (c) How much time is needed for them to complete one cycle?The electric field of an electromagnetic wave traveling in vacuum is described by the following wave function: E =(5.00V/m)cos[kx(6.00109s1)t+0.40] j where k is the wavenumber in rad/m, x is in m, t s in Find the following quantities: (a) amplitude (b) frequency (c) wavelength (d) the direction of the travel of the wave (e) the associated magnetic field wave A plane electromagnetic wave of frequency 20 GHz moves in the positive y-axis direction such that its electric field is pointed along the z-axis. The amplitude of the electric field is 10 V/m. The start of time is chosen so that at t = 0, the electric field has a value 10 V/m at the origin. (a) Write the wave function that will describe the electric field wave, (b) Find the wave function that will describe the associated magnetic field wave.The following represents an electromagnetic wave traveling in the direction of the positive y-axis: Ex=0;E0cos(kxt);Ez=0Bx=0;By=0;Bz=B0cos(kxt) The wave is passing through a wide tube of circular cross- section of radius R whose axis is along they-axis. Find the expression for the displacement current through the tube.While outdoors on a sunny day, a student holds a large convex lens of radius 4.0 cm above a sheet of paper to produce a bright spot on the paper that Is 1.0 cm in radius, rather than a sharp focus. By what factor is the electric field in the bright spot of light related to the electric field of sunlight leaving the side of the lens facing the paper?A plane electromagnetic wave travels northward. At one instant, its electric field has a magnitude of 6.0 V/m and points eastward. What are the magnitude and direction of the magnetic field at this instant?The electric field of an electromagnetic wave is given by E=(6.0103V/m)sin[2(x18mt6.0 10 8s)] j Write the equations for the associated magnetic field and Poynting vector.A radio station broadcasts at a frequency of 760 kHz. At a receiver some distance from the antenna, the maximum magnetic field of the electromagnetic wave detected is 2.151011T . (a) What is the maximum electric field? (b) What is the wavelength of the electromagnetic wave?The filament in a clear incandescent light bulb radiates visible light at a power of 5.00 W. Model the glass part of the bulb as a sphere of radius r0=3.00cm and calculate the amount of electromagnetic energy from visible light inside the bulb.At what distance does a 100-W lightbulb produce the same intensity of light as a 75-W lightbulb produces 10 m away? (Assume both have the same efficiency for converting electrical energy in the circuit into emitted electromagnetic energy.)An incandescent light bulb emits only 2.6 W of its power as visible light. What is the rms electric field of the emitted light at a distance of 3.0 m from the bulb?A 150-W lightbulb emits 5% of its energy as electromagnetic radiation. What is the magnitude of the average Poynting vector 10 m from the bulb?A small helium-neon laser has a power output of 2.5 mW What is the electromagnetic energy in a 1.0-m length of the beam?At the top of Earth’s atmosphere, the time-averaged Porting vector associated with sunlight has a magnitude of about 1.4W/m2 . (a) What are the maximum values of the electric and magnetic fields for a wave of this intensity? (b) What is the total power radiated by the sun? Assume that the Earth is 1.51011m from the Sun and that sunlight is composed of electromagnetic plane waves.The magnetic field of a plane electromagnetic wave moving along the z axis is given by B =B0(coskz+t) j , where B0=5.001010 and k=3.10102m-1 . (a) Write an expression for the electric field associated with the wave. (b) What are the frequency and the wavelength of the wave? (C) What is its average Poynting vector?What is the intensity of an electromagnetic wave with a peak electric field strength of 125 Vim?Assume the helium-neon lasers commonly used in student physics laboratories have power outputs of 0.500 mW. (a) If such a laser beam is projected onto a circular spot 1.00 mm in diameter, what Is its Intensity? (b) Find the peak magnetic field strength. (C) Find the peak electric field strength.An AM radio transmitter broadcasts 50.0 kW of power uniformly in all directions. (a) Assuming all of the radio waves that strike the ground are completely absorbed, and that there is no absorption by the atmosphere or other objects, what Is the intensity 30.0 km away? (Hint: Half the power will be spread over the area of a hemisphere.) (b) What is the maximum electric field strength at this distance?Suppose the maximum safe intensity of microwaves for human exposure is taken to be 1,00 W/m2. (a) If a radar unit leaks 10.0 W of microwaves (other than those sent by its antenna) uniformly in all directions, how far away must you be to be exposed to an intensity considered to be safe? Assume that the power spreads uniformly over the area of a sphere with no complications from absorption or reflection, (b) What is the maximum electric field strength at the safe intensity? (Note that early radar units leaked more than modem ones do. This caused identifiable health problems, such as cataracts, for people who worked near them.)A 2.50-rn-diameter university communications satellite dish receives TV sigia1s that have a maximum electric field strength (for one channel) of 7.50V/m (see below). (a) What is the intensity of this wave? (b) What is the power received by the antenna? (C) If the orbiting satellite broadcasts uniformly over an area of 1.501013m2 (a large fraction of North America), how much power does it radiate?Lasers can be constructed that produce an extremely high intensity electromagnetic wave for a brief time—called pulsed lasers. They are used to initiate nuclear fusion, for example. Such a laser may produce an electromagnetic wave with a maximum electric field strength of 1.001011 V/m for a time of 1.00 ns. (a) What is the maximum magnetic field strength in the wave? (b) What is the intensity of the beam? (c) What energy does it deliver on an 1.00-mm2 area?A 1-W lightbulb emits 5% of its energy as electromagnetic radiation. What is the radiation pressure on an absorbing sphere of radius 10 m that surrounds the bulb?What pressure does light emitted uniformly in all directions from a 100-W incandescent light bulb exert on a mirror at a distance of 3.0 in, if 2.6 W of the power is emitted as visible light?A microscopic spherical dust particle of radius 2m and mass 10g is moving in outer space at a constant speed of 30 cm/sec. A wave of light strikes it from the opposite direction of its motion and gets absorbed. Assuming the particle decelerates uniformly to zero speed in one second, what is the average electric field amplitude in the light?A Styrofoam spherical ball of radius 2 mm and mass 20g is to be suspended by the radiation pressure in a vacuum tube in a lab. How much intensity will be required if the light is completely absorbed the ball?Suppose that S avg for sunlight at a point on the surface of Earth is 900W/m2. (a) If sunlight falls perpendicularly on a kite with a reflecting surface of area 0.75m2 , what is the average force on the kite due to radiation pressure? (b) How is your answer affected if the kite material is black and absorbs all sunlight?reaches the ground with an intensity of about 1.0kW/m2 . A sunbather has a body surface area of 0.8 m2 facing the sun while reclining on a beach chair on a clear day. (a) how much energy from direct sunlight reaches the sunbather’s skin per second? (b) What pressure does the sunlight exert if it is absorbed?Suppose a spherical particle of mass m and radius R in space absorbs light of intensity I for time t. (a) How much work does the radiation pressure do to accelerate the particle (mm rest In the given tine It absorbs the light? (b) How much energy canted by the electromagnetic waves is absorbed by the particle over this time based on the radiant energy incident on the particle?How many helium atoms, each with a radius of about 31 pm, must be placed end to end to have a length equal to one wavelength of 470 nm blue light?If you wish to detect details of the size of atoms (about 0.2 nm) with electromagnetic radiation, it must have a wavelength of about this size. (a) What is its frequency? (b) What type of electromagnetic radiation might this be?Find the frequency range of visible light, given that it encompasses wavelengths from 380 to 760 nm.(a) Calculate the wavelength range for AM radio given its frequency range Is 540 to 1600 kHz. (b) Do the same for the FM frequency range of 88.0 to 108 MHz.Radio station WWVB, operated by the National Institute of Standards and Technology (NIST) from Fort Collins, Colorado, at a low frequency of 60 kHz, broadcasts a time synchronization signal whose range covers the entire continental US. The timing of the synchronization signal is controlled by a set of atomic clocks to an accuracy of 101012 s, and repeats every 1 minute. The signal is used for devices, such as radio-controlled watches, that automatically synchronize with it at preset local times. WWVB's long wavelength signal tends to propagate close to the ground. (a) Calculate the wavelength of the radio waves from WWVB. (b) Estimate the error that the travel time of the signal causes in synchronizing a radio controlled watch in Norfolk, Virginia, which is 1570 mi (2527 km) from Fort Collins, Colorado.An outdoor WIFi unit for a picnic area has a 100-mW output and a range of about 30 m. What output power would reduce its range to 12 m for use with the same devices as before? Assume there are no obstacles in the way and that microwaves into the ground are simply absorbed.The prefix “mega” (M) and “kilo” (k), when referring to amounts of computer data, refer to factors of 1024 or 210 rather than 1000 for the prefix kilo, and 10242 = 220 rather than 1,000,000 for the prefix Mega (M). If a wireless (WiFi) router transfers 150 Mbps of data, how many bits per second is that in decimal arithmetic?A computer user finds that his wireless router transmits data at a rate of 75 Mbps (megabits per second). Compare the average time to transmit one bit of data with the time difference between the wifi signal reaching an observer’s cell phone directly and by bouncing back to the observer from a wall 8.00 m past the observer.(a) The ideal size (most efficient) for a broadcast antenna with one end on the ground is one-fourth the wavelength (/4) of the electromagnetic radiation being sent out. If a new radio station has such an antenna that is 50.0 m high, what frequency does it broadcast most efficiently? Is this in the AM or FM band? (b) Discuss the analogy of the fundamental resonant mode of an air column closed at one end to the resonance of currents on an antenna that is one-fourth their wavelength.What are the wavelengths of (a) X-rays of frequency 2.01017 Hz? (b) Yellow light of frequency 5.11014Hz ? (C) Gamma rays of frequency 1.01023Hz ?For red light of =660nm , what are f,, and k?A radio transmitter broadcasts plane electromagnetic waves whose maximum electric field at a particular location is 1.55103V/m . What is the maximum magnitude of the oscillating magnetic field at that location? How does it compare with Earth’s magnetic field?(a) Two microwave frequencies authorized for use in microwave ovens are: 915 and 2450 MHz. Calculate the wavelength of each. (b) Which frequency would produce smaller hot spots in foods due to interference effects?During normal beating, the heart creates a maximum 4.00-mV potential across 0.300 m of a person’s chest, creating a 1.00-Hz electromagnetic wave. (a) What is the maximum electric field strength created? (b) What is the corresponding maximum magnetic field strength in the electromagnetic wave? (C) What is the wavelength of the electromagnetic wave?Distances in space are often quoted in units of light years, the distance ugh travels in 1 year. (a) How many meters is a light-year? (b) How many meters is it to Andromeda, the nearest large galaxy, given that it is 2.54106 ly away’ (C) The most distant galaxy yet discovered is 13.4109 ly away. How far is this in meters?A certain 60.0-Hz ac power line radiates an electromagnetic wave having a maximum electric field strength of 13.0 kV/m. (a) What is the wavelength of this very-low-frequency electromagnetic wave? (b) What type of electromagnetic radiation is this wave (b) What is its maximum magnetic field strength?(a) What is the frequency of the 193-nm ultraviolet radiation used in laser eye surgery? (b) Assuming the accuracy with which this electromagnetic radiation can ablate (reshape) the cornea is directly proportional to wavelength, how much more accurate can this UV radiation be than the shortest visible wavelength of light?In a region of space, the electric field is pointed along the x-axis, but its magnitude changes as described by Ex=(10N/C)sin(20x500t)Ey=Ez=0 where t is in nanoseconds and x is in cm. Find the displacement current through a circle of radius 3 cm in the x = 0 plane at I = 0.A microwave oven uses electromagnetic waves of frequency f=2.45109 to heat foods. The waves reflect from the inside walls of the oven to produce an interference pattern of standing waves whose antinodes are hot spots that can leave observable pit marks in some foods. The pit marks are measured to be 6.0 cm apart. Use the method employed by Heinrich Hertz to calculate the speed of electromagnetic waves this implies.Galileo proposed measuring the speed of light by uncovering a lantern and having an assistant a known distance away uncover his lantern en he saw the light from Galileo’s lantern, and timing the delay. How far away must the assistant be for the delay to equal the human reaction time of about 0.25 s?Show that the wave equation in one dimension 2fx2=1v22ft2 Is satisfied by any doubly differentiable function of either the form f(x — vt) or f(x + vt).On its highest power setting, a microwave oven increases the temperature of 0.400 kg of spaghetti by 45.0°C in 120 s. (a) What was the rate of energy absorption by the spaghetti, given that its specific heat is 3.76103J/kg.C ? Assume the spaghetti is perfectly absorbing. (b) Find the average intensity of the microwaves, given that they are absorbed over a circular area 20.0 cm in diameter. (C) What is the peak electric field strength of the microwave? (d) What is its peak magnetic field strength?A certain microwave oven projects 1.00 kW of microwaves onto a 30-cm-by-40-cm area. (a) What is its intensity in W /m2? (b) Calculate the maximum electric field strength E0in these waves. (c) What is the maximum magnetic field strength B0?E1ecmagnedc radiation from a 5.00-mW laser is concentrated on a 100-mm2 area. (a) What is the intensity in W/m2? (b) Suppose a 2.00-nC electric charge is in the beam. What is the maximum electric force it experiences? (C) If the electric charge moves at 400 m/s, what maximum magnetic force can it feel?A 200-turn flat coil of wire 30.0 cm in diameter acts as an antenna for FM radio at a frequency of 100 MHz. The magnetic field of the incoming electromagnetic wave is perpendicular to the coil and has a maximum strength of 1.001012T . (a) What power is incident on the coil? (b) What average emf is induced in the coil over one-fourth of a cycle? (c) If the radio receiver has an inductance of 2.50H, what capacitance must it have to resonate at 100 MHz?Suppose a source of electromagnetic waves radiates uniformly in all directions in empty space where there are no absorption or interference effects. (a) Show that the intensity is inversely proportional to r2, distance from the source squared. (b) Show that the magnitudes of the electric and magnetic fields are inversely proportional to r.A radio station broadcasts its radio waves with a power of 50,000 W. What would be the intensity of this signal if it is received on a planet orbiting Proxima Centuri, the closest star to our Sun, at 4.243 ly away?The Poynting vector describes a flow of energy whenever electric and magnetic fields are present. Consider a long cylindrical wire of radius r with a current I in the wire, with resistance R and voltage V. From the expressions for the electric field along the wire and the magnetic field around the wire, obtain the magnitude and direction of the Poynting vector at the surface. Show that it accounts for an energy flow into the wire from the fields around it that accounts for the Ohmic heating of the wire.The Sun’s energy strikes Earth at an intensity of 1.37kW/m2 . Assume as a mode! approximation that all of the light is absorbed. (Actually, about 30% of the light intensity is reflected out into space.) (a) Calculate the total force that the Sun’s radiation exerts on Earth. (b) Compare this to the force of gravity between the Sun and Earth. Earth’s mass is 5.9721024 kg.If a Lightsail spacecraft were sent on a Mars mission, by si1at ratio of the final force to the initial force would its propulsion be reduced when It reached Mars?Lunar astronauts placed a reflector on the Moon’s surface, off which a laser beam is periodically reflected. The distance to the Moon is calculated from the round-trip time. (a) To what accuracy in meters can the distance to the Moon be determined, If this time can be measured to 0.100 ns? (b) What percent accuracy is this, given the average distance to the Moon is 384,480 kin?Radar is used to determine distances to various objects by measuring the round-trip time for an echo from the object, (a) How far away is the planet Venus if the echo time is 1000 s? (b) What is the echo time for a car 75.0 m from a highway police radar unit? (c) How accurately (in nanoseconds) must you be able to measure the echo time to an airplane 12.0 km away to determine its distance within 10.0 m?Calculate the ratio of the highest to lowest frequencies of electromagnetic waves the eye can see, given the wavelength range of visible light is from 380 to 760 nm. (Note that the ratio of highest to lowest frequencies the ear can hear is 1000.)How does the wavelength of radio waves for an AM radio station broadcasting at 1030 KHz compare with the wavelength of the lowest audible sound waves (of 20 Hz). The speed of sound in air at 20 °C is about 343 m/s.A parallel-plate capacitor with plate separation d is connected to a source of emf that places a time-dependent voltage V(t) across its circular plates of radius r0and area (a) Write an expression for the time rate of change of energy inside the capacitor in terms of V(t) and dV(t)/ dt. (b) Assuming that V(t) is increasing with time, identify the directions of the elecuic field lines inside the capacitor and of the magnetic field lines at the edge of the region between the plates, and then the direction of the Poynting vector S at this location. (c) Obtain expressions for the time dependence of E(t), for B(t) from the displacement current, and for the magnitude of the Poynting vector at the edge of the region between the plates. (d) From S , obtain an expression In terms of ‘(t) and dV(t)/dt for the rate at which electromagnetic field energy the region between the plates. (e) Compare the results of pails (a) and (d) and explain the relationship between them.A particle of cosmic dust has a density =2.0g/cm3 , (a) Assuming the dust particles are spherical and light absorbing, and are at the same distance as Earth from the Sun, determine the particle size for which radiation pressure from sunlight is equal to the Sun's force of gravity on the dust particle, (b) Explain how the forces compare if the particle radius is smaller, (c) Explain what this implies about the sizes of dust particle likely to be present in the inner solar system compared with outside the Oort cloud.

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