16.5 The electromagnetic spectrum  (Page 7/19)

A certain microwave oven projects 1.00 kW of microwaves onto a 30-cm-by-40-cm area. (a) What is its intensity in ${\text{W/m}}^2$ ? (b) Calculate the maximum electric field strength _{$E_0$} in these waves. (c) What is the maximum magnetic field strength $B_0$ ?

Electromagnetic radiation from a 5.00-mW laser is concentrated on a $1.00{\text{-mm}}^2$ area. (a) What is the intensity in ${\text{W/m}}^2$ ? (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 $1.00\times {10}^{\mathrm{-12}}\text{T}$ . (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 $2.50\mu \text{H}$ , 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 $r^2$ , the 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 $1.37{\text{kW/m}}^2$ . Assume as a model 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 $5.972\times {10}^{24}\text{kg}.$

If a Lightsail spacecraft were sent on a Mars mission, by what fraction would its propulsion force 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 384,400 km?

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 $20\text{°C}$ 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 $r_0$ and area $A=\pi r_0^2$ (see below).

(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 electric 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 $\overrightarrow{S}$ 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 $\overrightarrow{S}$ , obtain an expression in terms of V ( t ) and dV ( t )/ dt for the rate at which electromagnetic field energy enters the region between the plates.

(e) Compare the results of parts (a) and (d) and explain the relationship between them.

A particle of cosmic dust has a density $\rho =2.0{\text{g/cm}}^3.$ (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.