Demonstrate the validity of the generalized form of Ampres

True or false: (a) The displacement current has different units than the conduction current. (b) Displacement current only exists if the electric field in the region is changing with time. (c) In an oscillating circuit, no displacement current exists between the capacitor plates when the capacitor is momentarily fully charged. (d) In an oscillating circuit, no displacement current exists between the capacitor plates when the capacitor is momentarily uncharged

Using SI units, show that has units of current

True or false: (a) Maxwells equations apply only to electric and magnetic fields that are constant over time. (b) The electromagnetic wave equation can be derived from Maxwells equations. (c) Electromagnetic waves are transverse waves. (d) The electric and magnetic fields of an electromagnetic wave in free space are in phase

Theorists have speculated about the existence of magnetic monopoles, and several experimental searches for such monopoles have occurred. Suppose magnetic monopoles were found and that the magnetic field at a distance from a monopole of strength is given by Modify the Gausss law for magnetism equation to be consistent with such a discovery

(a) For each of the following pairs of electromagnetic waves, which has the higher frequency: (1) visible light or X rays, (2) green light or red light, (3) infrared waves or red light. (b) For each of the following pairs of electromagnetic waves, which has the longer wavelength: (1) visible light or microwaves, (2) green light or ultraviolet light, (3) gamma rays or ultraviolet light

The detection of radio waves can be accomplished with either an electric dipole antenna or a loop antenna. True or false: (a) The electric dipole antenna works according to the Faradays law. (b) If a linearly polarized radio wave is approaching you head on such that its electric field oscillates vertically, to best detect this wave the normal to a loop antennas plane should be oriented so that it points either right or left. (c) If a linearly polarized radio wave is approaching you such that its electric field oscillates in a horizontal plane, to best detect this wave using a dipole antenna the antenna should be oriented vertically

Atransmitter emits electromagnetic waves using an electric dipole antenna oriented vertically. (a) A receiver to detect the waves also uses an electric dipole antenna that is one mile from the transmitting antenna and at the same altitude. How should the receivers electric dipole antenna be oriented for optimum signal reception? (b) A receiver to detect these waves uses a loop antenna that is one mile from the transmitting antenna and at the same altitude. How should the loop antenna be oriented for optimum signal reception

Show that the expression for the Poynting vector (Equation 30-21) has units of watts per square meter (the SI units for electromagnetic wave intensity)

If a red light beam, a green light beam, and a violet light beam, all traveling in empty space, have the same intensity, which light beam carries more momentum? (a) the red light beam, (b) the green light beam, (c) the violet light beam, (d) They all have the same momentum. (e) You cannot determine which beam carries the most momentum from the data given.

If a red light plane wave, a green light plane wave, and a violet light plane wave, all traveling in empty space, have the same intensity, which wave has the largest peak electric field? (a) the red light wave, (b) the green light wave, (c) the violet light wave, (d) They all have the same peak electric field. (e) You cannot determine the largest peak electric field from the data given.

Two sinusoidal plane electromagnetic waves are identical except that wave Ahas a peak electric field that is three times the peak electric field of wave B. How do their intensities compare? (a) (b) (c) (d) (e) You cannot determine how their intensities compare from the data given.

ENGINEERING APPLICATION In laser cooling and trapping, the forces associated with radiation pressure are used to slow down atoms from thermal speeds of hundreds of meters per second at room temperature to speeds of a few meters per second or slower. An isolated atom will absorb only radiation of specific frequencies. If the frequency of the laser-beam radiation is tuned so that the target atoms will absorb the radiation, then the radiation is absorbed during a process called resonant absorption. The cross-sectional area of the atom for resonant absorption is approximately equal to where is the wavelength of the laser light. (a) Estimate the acceleration of a rubidium atom (molar mass in a laser beam whose wavelength is and intensity is (b) About how long would it take such a light beam to slow a rubidium atom in a gas at room temperature to near-zero speed?

ENGINEERING APPLICATION One of the first successful satellites launched by the United States in the 1950s was essentially a large spherical (aluminized) Mylar balloon from which radio signals were reflected. After several orbits around Earth, scientists noticed that the orbit itself was changing with time. They eventually determined that radiation pressure from sunlight was causing the orbit of this object to changea phenomenon not taken into account in planning the mission. Estimate the ratio of the radiation-pressure force by the sunlight on the satellite to the gravitational force by Earths gravity on the satellite.

Some science fiction writers have described solar sails that could propel interstellar spaceships. Imagine a giant sail on a spacecraft subjected to radiation pressure from our Sun. (a) Explain why this arrangement works better if the sail is highly reflective rather than highly absorptive. (b) If the sail is assumed highly reflective, show that the force exerted by the sunlight on the spacecraft’s sail is given by \(P_S A/(4 \pi r^2 c)\) where \(P_S\) is the power output of the Sun \((3.8 \times 10^{26}~ \mathrm W)\) is the surface area of the sail, is the distance from the Sun, and is the speed of light. (Assume the area of the sail is much larger than the area of the spacecraft so that all the force is due to radiation pressure on the sail only.) (c) Using a reasonable value for compare the force on the spacecraft due to the radiation pressure and the force on the spacecraft due to the gravitational force of the Sun on the spacecraft. Does the result imply that such a system will work? Explain your answer.

A parallel-plate capacitor has circular plates and no dielectric between the plates. Each plate has a radius equal to and the plates are separated by 1.1 mm. Charge is flowing onto the upper plate (and off of the lower plate) at a rate of (a) Find the rate of change of the electric field strength in the region between the plates. (b) Compute the displacement current in the region between the plates and show that it equals 5.0 A

In a region of space, the electric field varies with time as where Find the peak displacement current through a surface that is perpendicular to the electric field and has an area equal to

For Problem 15, show that the magnetic field strength between the plates a distance from the axis through the centers of both plates is given by

The capacitors referred to in this problem have only empty space between the plates. (a) Show that a parallel-plate capacitor has a displacement current in the region between its plates that is given by where is the capacitance and is the potential difference between the plates. (b) A parallelplate capacitor is connected to an ideal ac generator so the potential difference between the plates is given by where and Find the displacement current in the region between the plates as a function of time.

There is a current of in a resistor that is connected in series with a parallel-plate capacitor. The plates of the capacitor have an area of and no dielectric exists between the plates. (a) What is the displacement current between the plates? (b) What is the rate of change of the electric field strength between the plates? (c) Find the value of the line integral where the integration path is a circle that lies in a plane that is parallel with the plates and is completely within the region between them.

Demonstrate the validity of the generalized form of Ampère's law (Equation 30-4) by showing that it gives the same result as the Biot-Savart law (Equation 27-3) in a specified situation. Figure 30-13 shows two momentarily equal but opposite point charges (+Q and -Q) on the x axis at x=-a and x=+a, respectively. At the same instant there is a current I in the wire connecting them, as shown. Point P is on the y axis at y=R. (a) Use the Biot-Savart law to show that the magnitude of the magnetic field at point P is given by \(B=\frac{\mu_0 I a}{2 \pi R} \frac{1}{\sqrt{R^2+a^2}}\). (b) Now consider a circular strip of radius r and width dr in the x=0 plane that has its center at the origin. Show that the flux of the electric field through this strip is given by \(E_x d A=\frac{Q}{\epsilon_0\left(r^2+a^2\right)^{3 / 2}} \pi r d r\). (c) Use the result from Part (b) to show that the total electric flux \(\phi_{\mathrm{e}}\) through a circular surface S of radius R is given by \(\phi_{\mathrm{e}}=\frac{Q}{\epsilon_{\mathrm{o}}}\left(1-\frac{a}{\sqrt{a^2+R^2}}\right)\) (d) Find the displacement current \(I_{\mathrm{d}} through S, and show that \(I+I_{\mathrm{d}}=I \frac{a}{\sqrt{a^2+R^2}}\). (e) Finally, show that the generalized form of Ampère's law (Equation 30-4 ) gives the same result for the magnetic field as found in Part (a).

The color of the dominant light from the Sun is in the yellow-green region of the visible spectrum. Estimate the wavelength and frequency of the dominant light emitted by our Sun. Hint: See Table 30-1.

(a) What is the frequency of microwave radiation that has a wavelength? (b) Using Table 30-1, estimate the ratio of the shortest wavelength of green light to the shortest wavelength of red light.

(a) What is the frequency of an X ray that has a wavelength? (b) The human eye is sensitive to light that has a wavelength equal to What are the color and frequency of this light? Comment on how this answer compares to your answer for Problem 21.

Suppose a radiating electric dipole lies along the axis. Let be the intensity of the radiation at a distance of and at angle of Find the intensity (in terms of at (a) a distance of and an angle of (b) a distance of and an angle of and (c) a distance of and an angle of

(a) For the situation described in Problem 24, at what angle is the intensity at a distance of equal to (b) At what distance is the intensity equal to when

ENGINEERING APPLICATION, CONTEXT-RICH You and your engineering crew are in charge of setting up a wireless telephone network for a village in a mountainous region. The transmitting antenna of one station is an electric dipole antenna located atop a mountain above sea level. There is a nearby mountain that is from the antenna and is also above sea level. At that location, one member of the crew measures the intensity of the signal to be What should be the intensity of the signal at the village that is located at sea level and from the transmitter?

ENGINEERING APPLICATION Aradio station that uses a vertical electric dipole antenna broadcasts at a frequency of and has a total power output of Calculate the intensity of the signal at a horizontal distance of from the station.

ENGINEERING APPLICATION Regulations require that licensed radio stations have limits on their broadcast power so as to avoid interference with signals from distant stations. You are in charge of checking compliance with the law. At a distance of from a radio station that broadcasts from a single vertical electric dipole antenna at a frequency of the intensity of the electromagnetic wave is What is the total power radiated by the station?

ENGINEERING APPLICATION A small private plane approaching an airport is flying at an altitude of above sea level. As a flight controller at the airport, you know your system uses a vertical electric dipole antenna to transmit at What is the intensity of the signal at the planes receiving antenna when the plane is 4.00 km from the airport? Assume the airport is at sea level.

An electromagnetic wave has an intensity of Find its (a) rms electric field strength, and (b) rms magnetic field strength.

The amplitude of an electromagnetic waves electric field is Find the waves (a) rms electric field strength, (b) rms magnetic field strength, (c) intensity, and (d) radiation pressure

The rms value of an electromagnetic waves electric field strength is Find the waves (a) rms magnetic field strength, (b) average energy density, and (c) intensity.

(a) An electromagnetic wave that has an intensity equal to is normal to a black by rectangular card that absorbs 100 percent of the wave. Find the force exerted on the card by the radiation. (b) Find the force exerted by the same wave if the card reflects 100 percent of the wave.

Find the force exerted by the electromagnetic wave on the card in Part (b) of Problem 33 if both the incident and the reflected waves are at angles of to the normal.

(a) For a given distance from a radiating electric dipole, at what angle (expressed as and measured from the dipole axis) is the intensity equal to 50 percent of the maximum intensity? (b) At what angle is the intensity equal to 1 percent of the maximum intensity?

A laser pulse has an energy of and a beam radius of The pulse duration is and the energy density is uniformly distributed within the pulse. (a) What is the spatial length of the pulse? (b) What is the energy density within the pulse? (c) Find the rms values of the electric and magnetic fields in the pulse.

An electromagnetic plane wave has an electric field that is parallel to the y axis, and has a Poynting vector given by \(\vec{S}(x, t)= \left(100 \mathrm{~W} / \mathrm{m}^2\right) \cos ^2(k x-\omega t) \hat{i}\), where x is in meters, \(k=10.0 \mathrm{rad} / \mathrm{m}\), \(\omega=3.00 \times 10^9 \mathrm{rad} / \mathrm{s}\), and t is in seconds. (a) What is the direction of propagation of the wave? (b) Find the wavelength and frequency of the wave. (c) Find the electric and magnetic fields of the wave as functions of x and t.

A parallel-plate capacitor is being charged. The capacitor consists of a pair of identical circular parallel plates that each have a radius and a separation distance (a) Show that the displacement current in the capacitor gap has the same value as the conduction current in the capacitor leads. (b) What is the direction of the Poynting vector in the region between the capacitor plates? (c) Find an expression for the Poynting vector in this region and show that its flux into the region between the plates is equal to the rate of change of the energy stored in the capacitor.

A pulsed laser fires a pulse that has a duration at a small object that has a mass equal to and is suspended by a fine fiber that is long. If the radiation is completely absorbed by the object, what is the maximum angle of deflection of this pendulum? (Think of the system as a ballistic pendulum and assume the small object was hanging vertically before the radiation hit it.)

The mirrors used in a particular type of laser are 99.99% reflecting. (a) If the laser has an average output power of what is the average power of the radiation incident on one of the mirrors? (b) What is the force due to radiation pressure on one of the mirrors?

(a) Estimate the force on Earth due to the pressure of the radiation on Earth by the Sun, and compare this force to the gravitational force of the Sun on Earth. (At Earths orbit, the intensity of sunlight is (b) Repeat Part (a) for Mars which is at an average distance of from the Sun and has a radius of (c) Which planet has the larger ratio of radiation pressure to gravitational attraction? SSM

Show by direct substitution that Equation 30-8a is satisfied by the wave function \(E_y=E_0 \sin (k x-\omega t)=E_0 \sin k(x-c t)\) where \(c=\omega / k\).

Use the values of and in SI units to compute and show that it is equal to

(a) Use Maxwells equations to show for a plane wave, in which and are independent of and that and (b) Show that and also satisfy the wave equation.

Show that any function of the form or satisfies the wave equation (Equation 30-7).

An electromagnetic wave has a frequency of and is traveling in a vacuum. The magnetic field is given by (a) Find the wavelength and the direction of propagation of this wave. (b) Find the electric field vector (c) Determine the Poynting vector, and use it to find the intensity of the wave.

ENGINEERING APPLICATION A circular loop of wire can be used to detect electromagnetic waves. Suppose the signal strength from a FM radio station distant is and suppose the signal is vertically polarized. What is the maximum rms voltage induced in your antenna, assuming your antenna is a loop?

ENGINEERING APPLICATION The electric field strength from a radio station some distance from the electric dipole transmitting antenna is given by (a) What peak voltage is picked up on a wire oriented parallel with the electric field direction? (b) What is the maximum voltage that can be induced by this electromagnetic wave in a conducting loop of radius and what orientation of the loop does this require?

Aparallel-plate capacitor has circular plates of radius that are separated by a distance In the gap between the two plates is a thin straight wire of resistance that connects the centers of the two plates. A time-varying voltage given by is applied across the plates. (a) What is the current drawn by the capacitor? (b) What is the magnetic field as a function of the radial distance from the centerline within the capacitor plates? (c) What is the phase angle between the current drawn by the capacitor and the applied voltage?

A beam of electromagnetic radiation is normal to a surface that reflects 50 percent of the radiation. What is the force exerted by the radiation on the surface?

The electric fields of two harmonic electromagnetic waves of angular frequency and are given by and by For the resultant of these two waves, find (a) the instantaneous Poynting vector and (b) the time-averaged Poynting vector. (c) Repeat Parts (a) and (b) if the direction of propagation of the second wave is reversed so that

Show that (Equation 30-10) follows from (Equation 30-6d and where I _ 0) by integrating along a suitable curve and over a suitable surface in a manner that parallels the derivation of Equation 30-9.

For your backpacking excursions, you have purchased a radio capable of detecting a signal as weak as This radio has a coil antenna that has a radius of wound on an iron core that increases the magnetic field by a factor of 200. The broadcast frequency of the radio station is (a) What is the peak magnetic field strength of an electromagnetic wave of this minimum intensity? (b) What is the peak emf that it is capable of inducing in the antenna? (c) What would be the peak emf induced in a straight metal wire oriented parallel to the direction of the electric field?

The intensity of the sunlight striking Earths upper atmosphere is (a) Find the rms values of the magnetic and electric fields of this light. (b) Find the average power output of the Sun. (c) Find the intensity and the radiation pressure at the surface of the Sun.

A conductor in the shape of a long solid cylinder that has a length a radius and a resistivity carries a steady current that is uniformly distributed over its cross section. (a) Use Ohms law to relate the electric field in the conductor to and (b) Find the magnetic field just outside the conductor. (c) Use the results from Part (a) and Part (b) to compute the Poynting vector at (the edge of the conductor). In what direction is (d) Find the flux through the surface of the cylinder, and use the flux to show that the rate of energy flow into the conductor equals where is the resistance of the cylinder.

A long solenoid that has turns per unit length carries a current that increases linearly with time. The solenoid has radius length and the current in the windings is given by (a) Find the induced electric field at a distance from the central axis of the solenoid. (b) Find the magnitude and direction of the Poynting vector at (just inside the solenoid windings). (c) Calculate the flux into the region inside the solenoid, and show that the flux equals the rate of increase of the magnetic energy inside the solenoid.

Small particles are blown out of the solar system by the radiation pressure of sunlight. Assume that each particle is spherical, has a radius has a density of and absorbs all the radiation in a cross-sectional area of Assume the particles are located at some distance from the Sun, which has a total power output of (a) What is the critical value for the radius of the particle for which the radiation force of repulsion just balances the gravitational force of attraction to the Sun? (b) Do particles that have radii larger than the critical value get ejected from the solar system, or is it only particles that have radii smaller than the critical value that get ejected? Explain your answer.

When an electromagnetic wave at normal incidence on a perfectly conducting surface is reflected, the electric field of the reflected wave at the reflecting surface is equal and opposite to the electric field of the incident wave at the reflecting surface. (a) Explain why this assertion is valid. (b) Show that the superposition of incident and reflected waves results in a standing wave. (c) Are the magnetic fields of the incident waves and reflected waves at the reflecting surface equal and opposite as well? Explain your answer.

An intense point source of light radiates isotropically (uniformly in all directions). The source is located above an infinite, perfectly reflecting plane. Determine the force that the radiation pressure exerts on the plane. SSM