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24.4 Energy in electromagnetic waves  (Page 3/6)

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Problems&Exercises

What is the intensity of an electromagnetic wave with a peak electric field strength of 125 V/m?

I = cε 0 E 0 2 2 = 3.00 × 10 8 m/s 8.85 × 10 –12 C 2 /N ⋅ m 2 1 25 V/m 2 2 = 20. 7 W/m 2

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Find the intensity of an electromagnetic wave having a peak magnetic field strength of 4 . 00 × 10 − 9 T size 12{4 "." "00"´"10" rSup { size 8{-9} } " T"} {} .

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Assume the helium-neon lasers commonly used in student physics laboratories have power outputs of 0.250 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.

(a) I = P A = P π r 2 = 0 . 250 × 10 − 3 W π 0 . 500 × 10 − 3 m 2 = 318 W/m 2 size 12{I= { {P} over {A} } = { {P} over {p r rSup { size 8{2} } } } = { {0 "." "250"´"10" rSup { size 8{-3} } " W"} over {∂ left (0 "." "500"´"10" rSup { size 8{-3} } " m" right ) rSup { size 8{2} } } } ="318 W/m" rSup { size 8{2} } } {}

(b) I ave = cB 0 2 2μ 0 ⇒ B 0 = 2μ 0 I c 1 / 2 = 2 4 π × 10 − 7 T ⋅ m/A 318 . 3 W/m 2 3.00 × 10 8 m/s 1 / 2 = 1 . 63 × 10 − 6 T alignl { stack { size 12{I rSub { size 8{"ave"} } = { { ital "cB" rSub { size 8{0} rSup { size 8{2} } } } over {2m rSub { size 8{0} } } } drarrow B rSub { size 8{0} } = left ( { {2m rSub { size 8{0} } I} over {c} } right ) rSup { size 8{1/2} } } {} #= left [ { {2 left (4¶´"10" rSup { size 8{-7} } " T" cdot "m/A" right ) left ("318" "." "3 W/m" rSup { size 8{2} } right )} over {3 "." "00"´"10" rSup { size 8{8} } " m/s"} } right ] rSup { size 8{ {1} slash {2} } } {} #= {underline {1 "." "63"´"10" rSup { size 8{-6} } " T"}} {} } } {}

(c) E 0 = cB 0 = 3 .00 × 10 8 m/s 1.633 × 10 − 6 T = 4 . 90 × 10 2 V/m alignl { stack { size 12{E rSub { size 8{0} } = ital "cB" rSub { size 8{0} } = left (3 "." "00"´"10" rSup { size 8{8} } " m/s" right ) left (1 "." "633"´"10" rSup { size 8{-6} } " T" right )} {} #= {underline {4 "." "90"´"10" rSup { size 8{2} } " V/m"}} {} } } {}

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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?

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Suppose the maximum safe intensity of microwaves for human exposure is taken to be 1 . 00 W /m 2 size 12{1 "." "00""W/m" rSup { size 8{2} } } {} . (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 modern ones do. This caused identifiable health problems, such as cataracts, for people who worked near them.)

(a) 89.2 cm

(b) 27.4 V/m

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A 2.50-m-diameter university communications satellite dish receives TV signals that have a maximum electric field strength (for one channel) of 7 . 50 μ V/m size 12{7 "." "50" mV/m} {} . (See [link] .) (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 . 50 × 10 13 m 2 size 12{1 "." "50"´"10" rSup { size 8{"13"} } " m" rSup { size 8{2} } } {} (a large fraction of North America), how much power does it radiate?

A large, round dish antenna looking like a giant white saucer is shown. It rests on a pillar like structure based on the ground. It is shown to receive TV signals in the form of electromagnetic waves shown as wavy arrows.
Satellite dishes receive TV signals sent from orbit. Although the signals are quite weak, the receiver can detect them by being tuned to resonate at their frequency.
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Lasers can be constructed that produce an extremely high intensity electromagnetic wave for a brief time—called pulsed lasers. They are used to ignite nuclear fusion, for example. Such a laser may produce an electromagnetic wave with a maximum electric field strength of 1 . 00 × 10 11 V / m size 12{1 "." "00"´"10" rSup { size 8{"11"} } " 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 a 1 . 00 -mm 2 size 12{1 "." "00""-mm" rSup { size 8{2} } } {} area?

(a) 333 T

(b) 1 . 33 × 10 19 W/m 2 size 12{1 "." "33"´"10" rSup { size 8{"19"} } "W/m" rSup { size 8{2} } } {}

(c) 13.3 kJ

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Show that for a continuous sinusoidal electromagnetic wave, the peak intensity is twice the average intensity ( I 0 = 2 I ave size 12{I rSub { size 8{0} } =2I rSub { size 8{"ave"} } } {} ), using either the fact that E 0 = 2 E rms size 12{E rSub { size 8{0} } = sqrt {2} E rSub { size 8{"rms"} } } {} , or B 0 = 2 B rms size 12{B rSub { size 8{0} } = sqrt {2} B rSub { size 8{"rms"} } } {} , where rms means average (actually root mean square, a type of average).

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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