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I ave = cB 0 2 0 , size 12{I rSub { size 8{"ave"} } = { { ital "cB" rSub { size 8{0} } rSup { size 8{2} } } over {2μ rSub { size 8{0} } } } } {}

where B 0 size 12{B rSub { size 8{0} } } {} is the maximum magnetic field strength.

One more expression for I ave size 12{I rSub { size 8{"ave"} } } {} in terms of both electric and magnetic field strengths is useful. Substituting the fact that c B 0 = E 0 size 12{c cdot B rSub { size 8{0} } =E rSub { size 8{0} } } {} , the previous expression becomes

I ave = E 0 B 0 0 . size 12{I rSub { size 8{"ave"} } = { {E rSub { size 8{0} } B rSub { size 8{0} } } over {2μ rSub { size 8{0} } } } } {}

Whichever of the three preceding equations is most convenient can be used, since they are really just different versions of the same principle: Energy in a wave is related to amplitude squared. Furthermore, since these equations are based on the assumption that the electromagnetic waves are sinusoidal, peak intensity is twice the average; that is, I 0 = 2 I ave size 12{I rSub { size 8{0} } =2I rSub { size 8{"ave"} } } {} .

Calculate microwave intensities and fields

On its highest power setting, a certain microwave oven projects 1.00 kW of microwaves onto a 30.0 by 40.0 cm area. (a) What is the intensity in W/m 2 size 12{"W/m" rSup { size 8{2} } } {} ? (b) Calculate the peak electric field strength E 0 size 12{E rSub { size 8{0} } } {} in these waves. (c) What is the peak magnetic field strength B 0 size 12{B rSub { size 8{0} } } {} ?

Strategy

In part (a), we can find intensity from its definition as power per unit area. Once the intensity is known, we can use the equations below to find the field strengths asked for in parts (b) and (c).

Solution for (a)

Entering the given power into the definition of intensity, and noting the area is 0.300 by 0.400 m, yields

I = P A = 1 . 00 kW 0 . 300 m × 0 . 400 m . size 12{I= { {P} over {A} } = { {1 "." "00"" kW"} over {0 "." "300 m"×0 "." "400 m"} } } {}

Here I = I ave size 12{I=I rSub { size 8{"ave"} } } {} , so that

I ave = 1000 W 0 . 120 m 2 = 8 . 33 × 10 3 W/m 2 . size 12{I rSub { size 8{"ave"} } = { {"1000"" W"} over {0 "." "120"" m" rSup { size 8{2} } } } =8 "." "33"×"10" rSup { size 8{3} } " W/m" rSup { size 8{2} } } {}

Note that the peak intensity is twice the average:

I 0 = 2 I ave = 1 . 67 × 10 4 W / m 2 . size 12{I rSub { size 8{0} } =2I rSub { size 8{"ave"} } =1 "." "67" times "10" rSup { size 8{4} } {W} slash {m rSup { size 8{2} } } } {}

Solution for (b)

To find E 0 size 12{E rSub { size 8{0} } } {} , we can rearrange the first equation given above for I ave size 12{I rSub { size 8{"ave"} } } {} to give

E 0 = 2 I ave 0 1/2 . size 12{E rSub { size 8{0} } = left ( { {2I rSub { size 8{"ave"} } } over {ce rSub { size 8{0} } } } right ) rSup { size 8{ {1}wideslash {2} } } } {}

Entering known values gives

E 0 = 2 ( 8 . 33 × 10 3 W/m 2 ) ( 3 . 00 × 10 8 m/s ) ( 8.85 × 10 12 C 2 / N m 2 ) = 2.51 × 10 3 V/m . alignl { stack { size 12{E rSub { size 8{0} } = sqrt { { {2 \( 8 "." "33"´"10" rSup { size 8{3} } " W/m" rSup { size 8{2} } \) } over { \( 3 "." "00"´"10" rSup { size 8{8} } " m/s" \) \( 8 "." "85"´"10" rSup { size 8{ +- 2} } C rSup { size 8{2} } /N cdot m rSup { size 8{2} } \) } } } } {} #=2 "." "51"´"10" rSup { size 8{3} } " V/m" "." {} } } {}

Solution for (c)

Perhaps the easiest way to find magnetic field strength, now that the electric field strength is known, is to use the relationship given by

B 0 = E 0 c . size 12{B rSub { size 8{0} } = { {E rSub { size 8{0} } } over {c} } } {}

Entering known values gives

B 0 = 2.51 × 10 3 V/m 3.0 × 10 8 m/s = 8.35 × 10 6 T . alignl { stack { size 12{B rSub { size 8{0} } = { {2 "." "51"´"10" rSup { size 8{3} } " V/m"} over {3 "." 0´"10" rSup { size 8{8} } " m/s"} } } {} #=8 "." "35"´"10" rSup { size 8{-6} } " T" "." {} } } {}

Discussion

As before, a relatively strong electric field is accompanied by a relatively weak magnetic field in an electromagnetic wave, since B = E / c size 12{B= {E} slash {c} } {} , and c size 12{c} {} is a large number.

Section summary

  • The energy carried by any wave is proportional to its amplitude squared. For electromagnetic waves, this means intensity can be expressed as
    I ave = 0 E 0 2 2 , size 12{I rSub { size 8{"ave"} } = { {ce rSub { size 8{0} } E rSub { size 8{0} } rSup { size 8{2} } } over {2} } } {}

    where I ave size 12{I rSub { size 8{"ave"} } } {} is the average intensity in W/m 2 size 12{"W/m" rSup { size 8{2} } } {} , and E 0 size 12{E rSub { size 8{0} } } {} is the maximum electric field strength of a continuous sinusoidal wave.

  • This can also be expressed in terms of the maximum magnetic field strength B 0 size 12{B rSub { size 8{0} } } {} as
    I ave = cB 0 2 0 size 12{I rSub { size 8{"ave"} } = { { ital "cB" rSub { size 8{0} } rSup { size 8{2} } } over {2m rSub { size 8{0} } } } } {}

    and in terms of both electric and magnetic fields as

    I ave = E 0 B 0 0 . size 12{I rSub { size 8{"ave"} } = { {E rSub { size 8{0} } B rSub { size 8{0} } } over {2m rSub { size 8{0} } } } } {}
  • The three expressions for I ave size 12{I rSub { size 8{"ave"} } } {} are all equivalent.

Problems&Exercises

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

I = 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

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"} {} .

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 0 B 0 = 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"}} {} } } {}

Questions & Answers

more explanation on GDP
Isaac Reply
it is a country total out put of goods and services divided by the total population of the country.I think it can also be derived from the country labour force,,because it mostly depend on the labour force and the level of technology .
Tantoh
labour force and technological progress leads to greater production increases the GDP
Ahmed
What is economics?
Bubu Reply
by this time
Emmanuel
It is a social science that analyses production,distribution and consumption of goods and services
Emmanuel
A social science that study human behavior in relationship with decision making
Jessica
What are the typical patterns of GDP for a high-income economy like the United States in the long run and the short run?
mwangala Reply
What are the limitation and significant of macroeconomic
Usman Reply
explain the significance of concerpt of opportunity cost in planning
Mwanaid Reply
what is meant by the price elasticity of demand?
Martine Reply
when price of a commodity increase it's demand contracts , and whe the price of a commodity decreases it's demand expands so the degree of change in demand in response to change in own price of the commodity is called PED . Ed = percentage change in quantity demanded / percentage change in price
shaswat
What are the limitations of macroeconomic and their segnificant
Usman Reply
Discuss the role of competition in stimulating economic growth?
Daniel Reply
competition stimulate economic growth because in such types of economy,they is no monopoly power every supplier will want to produce to meet customers choice which brings about quality production and attract invested and customers into such economy
Koka
competition creates Monopoly because of economy of scale. it's not antithesis but different side of same coin
toko
competition result in high economic growth since every firm will intend to provide quality services and products to meet customers needs and requirements unlike in Monopoly situation where a firm just provide what it want to resulting in large stock piles of unwanted products ,ie inefficiency, howev
Mark
microeconomics study part of the economy but macroeconomic study the whole economy
Olokun Reply
studying the whole economy, solving the problem of the economy and building up the economy
Olokun
micro means small while macro means large
Olokun
standard of living is the footsteps of an economy because it plays important role for country to have crucial view about their budget ,import and export
Olokun
it will be differ because economic agent will only take their views on some part of household
Olokun
can opportunity cost be zero
OBED Reply
how many types of transportation do we have
Jacob
yes. when a customer's purchasing power is high, he may have d ability to purchase all he needs, dt makes opportunity cost zero
George
please can give more explanation on this question
OBED
what are the factors production
PETER Reply
Labour capital entrepreneurs
Leta
Land,capital, labour,and the entrepreneur
Tantoh
I will like to know use of calculus in economics
JHUMA Reply
do they use it in economics?
Pranav
I want to know if I should take calculus or statistics and probability my senior year of highschool
Yahir
yes for example in monopolistic competitive market..... TR=TC* & THIS CALCULATED BY CHANGING( DERIVATIVE LAW) MR =MC ** WILL BE THE FORMULA THAT USE.
Leta
please in which topic in economic is the question coming from.
Tantoh
from PCF in economics
Leta
why is unitary proportional to responsiveness
Etim Reply
any tip for igcse economics exam?pls
Stacey Reply
well
The
What is a market
Divine Reply
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
price of the related goods 2 price of the given commodity 3 income of the consumer 4 taste and preference 5 expectation in the future price
John
pls the taste and preference
Nas
explain briefly
Nas
a consumer taste and preference commodity changes for a time the man becomes
John
sorry sorry
John
is when the price of a commodity becomes high and can't afford example Samsung instead of iPhone
John
consumers who have high intense for goods will purchase the goods even if the price of that commodity increases because he or she preferred that commodity.people will be prefer iphone as its price increase
Yussif
as usual bad taste of preference is when a consumer regrets from one commodity to another in terms of the price
John
thanks alot
Nas
you're welcome
John
#Preference; #Income #Test
Dereje
#price Of Commodity #Income #Taste #Preference
Dereje
#Market is The Place Where Buyers And Sellers Are Exchanging Their Goods And service. #
Dereje
difference between macro and micro economics
Lawrence
Microeconomic Study about individual consumers market But Macroeconomis Study General economic Process Such As #Aggregate Demand #Aggregate Supples #GDp= #GNp
Dereje
nice so can u run down a brife discussion on GDP
Lawrence
good
Chinex
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
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Source:  OpenStax, Yupparaj english program physics corresponding to thai physics book #3. OpenStax CNX. May 19, 2014 Download for free at http://legacy.cnx.org/content/col11657/1.1
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