62 Energy in Electromagnetic Waves

Every day we encounter electromagnetic waves carrying energy. Sunlight warms our skin, Wi-Fi signals transmit information throughout buildings, microwave ovens heat food, and medical imaging systems use carefully controlled electromagnetic radiation to diagnose disease. In healthcare, the amount of energy delivered by an electromagnetic wave is often just as important as its frequency. For example, a diagnostic X-ray must deliver enough energy to produce a clear image while minimizing the radiation dose to the patient. Likewise, lasers used during eye surgery must deliver energy precisely to the target tissue without damaging surrounding structures.

Electromagnetic waves transport energy through the oscillating electric and magnetic fields that make up the wave. These fields exert forces on charged particles, causing them to move and transferring energy to matter. When the frequency of the incoming wave matches a natural frequency of the material—as occurs when microwaves interact with water molecules or when radio-frequency pulses excite hydrogen nuclei during magnetic resonance imaging (MRI)—energy transfer becomes especially efficient.

Once generated, an electromagnetic wave carries energy away from its source whether or not it is immediately absorbed. If the wave encounters matter, some or all of its energy may be transferred to the material, causing heating, electrical currents, or atomic and molecular transitions. Any remaining energy continues traveling with the wave.

Just as for mechanical waves such as water waves or sound waves, the energy transported by an electromagnetic wave is proportional to the square of its amplitude. For electromagnetic waves, the amplitude corresponds to the maximum values of the electric field and magnetic field. Consequently, stronger electric and magnetic fields transport significantly more energy than weaker fields.

This relationship explains why a laser with twice the electric field strength does not simply deliver twice the energy—it delivers approximately four times as much energy. Similar principles apply in medicine, where increasing the intensity of a therapeutic laser or X-ray beam can dramatically increase the amount of energy deposited in tissue.

The quantity used to describe how rapidly electromagnetic energy is delivered is the intensity, denoted by I. Intensity is defined as the power delivered per unit area and is measured in watts per square meter ([latex]\text{W/m}^2[/latex]). For a continuous sinusoidal electromagnetic wave, the average intensity is

where

• [latex]c[/latex] is the speed of light,
• [latex]\epsilon_0[/latex] is the permittivity of free space, and
• [latex]E_0[/latex] is the maximum electric field strength.

Since the electric and magnetic fields in an electromagnetic wave are related by [latex]E_0=cB_0[/latex], the same average intensity can also be written in terms of the magnetic field strength:

where [latex]\mu_0[/latex] is the permeability of free space.

Substituting [latex]E_0=cB_0[/latex] into either expression produces a third equivalent equation involving both fields:

These three equations are simply different ways of expressing the same physical principle: the energy carried by an electromagnetic wave depends on the square of its amplitude. Depending on which quantities are known, any one of the equations may be the most convenient to use.

Finally, because these equations describe sinusoidal electromagnetic waves, the maximum (or peak) intensity is twice the average intensity:

Throughout this chapter, these relationships will allow us to calculate the intensity and field strengths of electromagnetic waves used in technologies ranging from microwave ovens to medical imaging systems and therapeutic lasers.

Problems & Exercises

30. What is the intensity of an electromagnetic wave with a peak electric field strength of 125 V/m?
31. Find the intensity of an electromagnetic wave having a peak magnetic field strength of [latex]4.00\times10^{-9}\ \text{T}[/latex].
32. Assume that a helium-neon laser commonly used in a student physics laboratory has a power output of 0.250 mW.
    1. If the laser beam is projected onto a circular spot 1.00 mm in diameter, what is its intensity?
    2. Find the peak magnetic field strength.
    3. Find the peak electric field strength.
33. An AM radio transmitter broadcasts 50.0 kW of power uniformly in all directions.
    1. Assuming that all the radio waves striking 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? Half of the transmitted power is spread over the area of a hemisphere.
    2. What is the maximum electric field strength at this distance?
34. Suppose the maximum acceptable intensity of microwaves for a particular exposure is taken to be [latex]1.00\ \text{W/m}^2[/latex]. A radar unit unintentionally leaks 10.0 W of microwave power uniformly in all directions.
    1. How far from the radar unit must a person be for the intensity to equal [latex]1.00\ \text{W/m}^2[/latex]? Assume that the power spreads uniformly over a sphere and that there is no absorption or reflection.
    2. What is the maximum electric field strength at this intensity?
35. A 2.50-m-diameter satellite dish receives a television signal with a maximum electric field strength of [latex]7.50\ \mu\text{V/m}[/latex], as shown in Figure 62.2.
    1. What is the intensity of the electromagnetic wave?
    2. What power is received by the dish?
    3. If the satellite broadcasts uniformly over an area of [latex]1.50\times10^{13}\ \text{m}^2[/latex], how much total power does it radiate?

36. Pulsed lasers can produce electromagnetic waves with extremely high intensity for very short periods. Suppose a laser produces a maximum electric field strength of [latex]1.00\times10^{11}\ \text{V/m}[/latex] for 1.00 ns.
    1. What is the maximum magnetic field strength?
    2. What is the average intensity of the beam?
    3. How much energy is delivered to an area of [latex]1.00\ \text{mm}^2[/latex]?
37. Show that, for a continuous sinusoidal electromagnetic wave, the peak intensity is twice the average intensity:
    [latex]I_0=2I_{\text{ave}}.[/latex]

    Use either [latex]E_0=\sqrt{2}E_{\text{rms}}[/latex] or [latex]B_0=\sqrt{2}B_{\text{rms}}[/latex].

38. Suppose a source radiates electromagnetic waves uniformly in all directions in empty space, with no absorption or interference.
    1. Show that the intensity is inversely proportional to the square of the distance from the source, [latex]I\propto1/r^2[/latex].
    2. Show that the magnitudes of the electric and magnetic fields are inversely proportional to distance, [latex]E_0\propto1/r[/latex] and [latex]B_0\propto1/r[/latex].
39. Integrated Concepts. An [latex]LC[/latex] circuit containing a 5.00-pF capacitor radiates electromagnetic waves with a wavelength of 3.30 m.
    1. What is the resonant frequency?
    2. What inductance is connected in series with the capacitor?
40. Integrated Concepts. What capacitance is needed in series with an [latex]800\ \mu\text{H}[/latex] inductor to form a circuit that radiates electromagnetic waves with a wavelength of 196 m?
41. Integrated Concepts. Police radar determines vehicle speed using a Doppler-shift technique similar to that used in medical ultrasound. The reflected signal experiences a double Doppler shift and is mixed with the original signal to produce beats. If [latex]1.50\times10^9\ \text{Hz}[/latex] microwaves produce a beat frequency of 150 Hz, what is the speed of the vehicle? Assume the usual Doppler-shift equations apply with the speed of sound replaced by the speed of light.
42. Integrated Concepts. Assume that the predominantly infrared radiation from a 200-W heat lamp behaves like a continuous electromagnetic wave with a wavelength of [latex]1.50\ \mu\text{m}[/latex]. The radiation is focused onto a circular area of a person's shoulder that is 25.0 cm in diameter.
    1. What is the intensity in [latex]\text{W/m}^2[/latex]?
    2. What is the peak electric field strength?
    3. What is the peak magnetic field strength?
    4. How long would it take to raise the temperature of a 4.00-kg portion of the shoulder by [latex]2.00^\circ\text{C}[/latex], assuming no other heat transfer and a specific heat of [latex]3.47\times10^3\ \text{J/(kg}\cdot^\circ\text{C)}[/latex]?
43. Integrated Concepts. A microwave oven increases the temperature of 0.400 kg of spaghetti by [latex]45.0^\circ\text{C}[/latex] in 120 s.
    1. What is the rate at which the spaghetti absorbs energy if its specific heat is [latex]3.76\times10^3\ \text{J/(kg}\cdot^\circ\text{C)}[/latex]?
    2. Find the average microwave intensity if the energy is absorbed over a circular area 20.0 cm in diameter.
    3. What is the peak electric field strength?
    4. What is the peak magnetic field strength?
44. Integrated Concepts. Electromagnetic radiation from a 5.00-mW laser is concentrated onto an area of [latex]1.00\ \text{mm}^2[/latex].
    1. What is the intensity in [latex]\text{W/m}^2[/latex]?
    2. Suppose a 2.00-nC charge is in the beam. What is the maximum electric force it experiences?
    3. If the charge moves at 400 m/s perpendicular to the magnetic field, what maximum magnetic force can it experience?
45. Integrated Concepts. A 200-turn flat coil, 30.0 cm in diameter, acts as an antenna for an FM radio signal with a frequency of 100 MHz. The magnetic field of the incoming wave is perpendicular to the coil and has a maximum strength of [latex]1.00\times10^{-12}\ \text{T}[/latex].
    1. What electromagnetic power is incident on the coil?
    2. What average emf is induced in the coil during one-fourth of a cycle?
    3. If the radio receiver has an inductance of [latex]2.50\ \mu\text{H}[/latex], what capacitance is required for resonance at 100 MHz?
46. Integrated Concepts. The electric and magnetic fields of an electromagnetic wave vary sinusoidally according to
    [latex]E=E_0\sin(2\pi ft)[/latex]

    and

    [latex]B=B_0\sin(2\pi ft).[/latex]

    Let [latex]f=1.00\ \text{GHz}[/latex].

    1. After [latex]t=0[/latex], when are the fields first zero again?
    2. When do they first reach their most negative values?
    3. How much time is required to complete one cycle?
47. Unreasonable Results. A researcher measures the wavelength of a 1.20-GHz electromagnetic wave to be 0.500 m.
    1. Calculate the speed at which the wave would propagate.
    2. What is unreasonable about this result?
    3. Which assumptions or measurements are inconsistent?
48. Unreasonable Results. The peak magnetic field strength in a residential microwave oven is reported to be [latex]9.20\times10^{-5}\ \text{T}[/latex].
    1. What intensity would this magnetic field imply?
    2. What is unreasonable about the result?
    3. What is likely wrong with the premise?
49. Unreasonable Results. An [latex]LC[/latex] circuit containing a 2.00-H inductor oscillates at a frequency that produces electromagnetic radiation with a wavelength of 1.00 m.
    1. What capacitance would the circuit require?
    2. What is unreasonable about the result?
    3. Which assumptions are inconsistent?
50. Unreasonable Results. An [latex]LC[/latex] circuit containing a 1.00-pF capacitor oscillates at a frequency that produces electromagnetic radiation with a wavelength of 300 nm.
    1. What inductance would the circuit require?
    2. What is unreasonable about the result?
    3. Which assumptions are inconsistent?
51. Create Your Own Problem. Consider electromagnetic fields produced by high-voltage power lines. Construct a problem in which the intensity of the electromagnetic radiation near a home is calculated from a measured magnetic field strength smaller than [latex]1\ \mu\text{T}[/latex]. Discuss how much electromagnetic power may be radiated by a section of power line several hundred meters long, and compare this value with the electrical power carried by the line.
52. Create Your Own Problem. Consider a residential satellite dish slightly less than 0.50 m in diameter. Construct a problem in which the power received by the dish and the maximum electric field strength of one microwave channel are calculated. Include reasonable values for the satellite's transmitted power, the area over which it spreads, and the collecting area of the dish.