Blackbody Spectrum SIM Homework Answer Key 1) In this question, you will use the Blackbody Spectrum Simulation to inesti!ate how the spectrum o" electroma!netic radiation emitted by ob#ects is a""ected by the ob#ect$s temperature% In this simulation, you can input the temperature and obsere the spectrum o" the radiation emitted% a) The temperature of stars in the universe varies with the type of star and the age of the star among other things. By looking at the shape of the spectrum of light emitted by a star, we can tell something about its average surface temperature. i) If we observe a star's spectrum and find that the peak power occurs at the border between red and infrared light, what is the approximate surface temperature of the star? in degrees !"
Using the Spectrum Simulation, when the temperature is 4040 Kelvin that peak power is at the border between the red and the infrared. Converting this to degrees Celsius: temperature in degrees C temperature in Kelvin ! "#$.%& temperature of the star 4040 K ! "#$.%& $#'# C ii) If we observe a stars spectrum and find that the peak power occurs at the border between blue and ultraviolet light, what is the surface temperature of the star? in degrees !"
Using the Spectrum Simulation, when the temperature is #%00 Kelvin that peak power is at the border between the blue and the ultraviolet. Converting this to degrees Celsius: temperature in degrees C temperature in Kelvin ! "#$.%& temperature of the star #%00 K ! "#$.%& '("# C b) #ight bulbs operate at $%&& degrees !. i) hat is the wavelength at which the most power is emitted for a light bulb operating at $%&& !?
)irst we must convert this "&00 C into Kelvin. temperature of bulb in Kelvin temperature in degrees C * "#$.%& temperature of bulb in Kelvin "&00 C * "#$.%& "##$ Kelvin Using the simulation we find the peak power is at %0&0 nm +nanometers. ii) (xplain why regular incandescent bulbs waste a lot of energy. Be sure to include your reasoning.
-he spectrum of power vs. wavelength at "&00 C shows that the maorit/ of the light emitted is at wavelengths longer than the visible. ll of the power +energ/ per second that goes into
producing light that is not at visible wavelengths is wasted power or energ/. Since onl/ about %"1 of the energ/ goes into producing visible light, regular incandescent bulbs are ver/ inefficient. c) Investigate how the observed spectrum responds to changing temperature. )ote you are only able to change temperature here, but should consider how the spectrum power vs. color" would change if you varied other characteristics about the ob*ect.
)alse If the only change you make is to decrease the temperature of True an ob*ect, the amount of power emitted at +&&& nm will increase in some cases. 2f the temperature of an obect decreases, the amount of power emitted at an/ wavelength will decrease. 2n the simulation, when the temperature is decreased ever/ point on the graph of power vs. color will lower indicating a reduction in power at ever/ color +or wavelength. -he peak of the curve will move toward the infrared end of the wavelength scale, but this peak is alwa/s lower than the peak power at higher temperatures. -rue
alse If the only change you make is to decrease the temperature of an ob*ect, the total amount of power emitted decreases in all cases.
3hen the temperature is decreased ever/ point on the graph of power vs. color will lower indicating a reduction in power at ever/ color +or wavelength. -hus, the total power must also have decreased. -otal radiation power emitted b/ an obect is given b/: emissivit/ 5 6olt7mann8s constant 5 +temperature 4 5 area of the surface 2f onl/ temperature changes, the total power goes up. )alse If you were to increase the surface area of a bulb filament, but True leave its temperature unchanged, then a larger fraction of its total power emitted would be emitted as I- radiation . -he temperature determines the shape of the spectrum and at what wavelength the peak power occurs. Changing the surface area of the bulb would increase the power emitted, but this increase would be seen at all wavelengths. 2f the surface are was increased b/ "&1, the power emitted at each wavelength would increase b/ "&1, increasing the total power emitted but 9- C;9<29< the fraction of the total power in the 2=.
d) se the /pectrum /imulation to investigate changes in the amount of light at visible wavelengths due to this change in temperature from $%&& ! to $&&& !. i) hat is the approximate ratio between the powers emitted at %&& nm at $&&& degrees ! to that at $%&& degrees !, that is, the power emitted at %&& nm at $&&& degrees ! divided by the amount of power at %&& nm at degrees $%&& !?
-he Spectrum simulation shows the amount of power a bulb emits at each wavelength. )or this problem, /ou needed to compare the curves for a "000 degree C + 2273 Kelvin bulb with that for at "&00 C +2773 Kelvin bulb. 3ith both curves displa/ed and 7oomed in on the wavelength scale, /ou find the &00 nm point on the wavelength +5!a5is and compare the relative heights of the curves +using the ruler to help /ou. 3e know to e5pect the power emitted at &00 nm when the bulb is "000 C to be less than that emitted when the bulb is "&00 C, so we know to e5pect a number less than %. Using the ruler to help, the ratio of the power emitted at &00 nm for "000 C to that for "&00 C is about %.% to %%. r =atio %.%>%% 0.%. ii) In 0uestion +b, we found the wavelength for the peak power of a bulb operating at $%&& degrees !. hat is the approximate peak wavelength for a bulb operating at $&&& degrees !?
Using the spectrum simulation, the peak wavelength for a bulb operating at "000 degrees C +""#$ Kelvin is about %"(0 nm +longer than the peak wavelength for a bulb operation at "&00 C e) 1ou turn up the dimmer switch so that the temperature of the filament reaches $2&& degrees !. The light bulb filament has an area of 2.3% x +&43 m$ &.+ s0uare inches" and an emissivity of &.5. 6ow much electrical power must it be using?
)rom conservation of energ/, we know that the electrical power being used must e?ual to the amount of power radiated b/ the filament if the temperature of the filament is stable. -he total power radiated b/ this bulb: emissivit/ 5 6olt7mann8s constant 5 +temperature 4 5 area of the surface. 0.( 5 +&.'# 5 %0!( @>+s m" K4 5 +"(#$ K4 5 '.4& 5 %0!4 m" 0.( 5 &.'# 5 %0!( 5 +"(#$ 4 5 '.4& 5 %0!4 @oules>s %AA$ 3atts
&) In this problem we will e'plore the !reenhouse e""ect by usin! the (reenhouse ""ect Simulation aailable "rom the 1*1* homepa!e% a) (xplore the simulation.
)alse The only effect of increasing the number of clouds is to True reduce the amount of sunlight absorbed by the surface of the earth. -he clouds do reflect sunlight back to space and reduce the amount of sunlight absorbed. -he/ also absorb the infrared radiation emitted b/ the surface and then reemit it either back towards the ground or towards space. )alse Increasing the concentration of greenhouse gases, increases True the amount of radiation that (arth emits to space. 2f there is a large change in greenhouse gas concentration, the Barth is not as good an emitter as it used to be. So if the amount of radiation emitted b/ the Barth to space used to e?ual the amount of radiation absorbed, now it will be less than the amount of radiation absorbed, because some of the radiation which previousl/ would have been emitted is absorbed b/ the atmosphere and re!emitted down to the ground. 9ow the Barth will take in more energ/ than it loses to space, and as a result the Barth will warm up. 3hen the Barth8s temperature is stable +no matter what the concentration of greenhouse gases and no matter what the surface temperature of the BarthD, the Barth is emitting the same amount of radiation to space that it absorbs from the sunlight: it is neither gaining nor losing energ/. 2n practice, the greenhouse gas concentrations change slowl/ and the Barth8s average temperature is stable, so the earth is alwa/s in radiative balance so the amount of radiation the Barth emits to space is constant ... alwa/s balancing the amount of radiation the earth absorbs for the sun. )alse hen sunlight encounters a cloud, the cloud reflects about True +&7 of the sunlight back to space. 3hen sunlight hits a cloud, the clouds reflect much more that %01 of the sunlight. 2t8s around &0!'01. -rue
alse hen there is a very large concentration of greenhouse gases, most of the I- radiation reaching space has interacted with greenhouse gas molecules
on its way from the surface to space.
-he greenhouse gases absorb 2= radiation. ;alf of the radiation is then redirected back down towards the ground and half towards space. -he higher the concentration of greenhouse gases, the more likel/ it is that the 2= radiation emitted b/ the surface will be intercepted b/ one of the greenhouse gas molecules. True )alse The total amount of radiation absorbed by the (arth8s surface is not affected by the concentration of greenhouse gases in the atmosphere.
3ithout greenhouse gases, the onl/ radiation absorbed b/ the Barth is sunlight. 6ut when the atmosphere has greenhouse gases, some of the 2= radiation emitted b/ the surface is absorbed and redirected back down to the surface. Under these conditions, the surface is absorbing both the sunlight radiation and 2= radiation, so the total amount of radiation absorbed increases. -rue
alse 9t higher temperatures, the (arth8s surface emits more I-
radiation.
s with an/ obect +bulbs, heaters, if the temperature increases the amount of power radiated increases. -rue
alse :uring the ice age, the amount of sunlight absorbed by the (arth8s surface decreased.
Euring the ice age, the ice reflected some sunlight back to space, so less was absorbed b/ the surface. True )alse 9ll greenhouse gases are from anthropogenic sources that is due to man8s activities".
-he maorit/ of the greenhouse gases are natural. 3ater vapor is the strongest greenhouse gas. Carbon dio5ide is naturall/ emitted b/ the plants although man8s activities have led to an increase in the concentration of carbon dio5ide in the atmosphere. b) (xplain your reasoning for your answer to the T; 0uestion
3ithout greenhouse gases, the onl/ radiation absorbed b/ the Barth is sunlight. 6ut when the atmosphere has greenhouse gases, some of the 2= radiation emitted b/ the surface is absorbed and redirected
back down to the surface. Under these conditions, the surface is absorbing both the sunlight radiation and 2= radiation, so the total amount of radiation absorbed increases. c) rom your observations of the simulation and your understanding of the basic physics principles of energy and radiation, explain why the average surface temperature of the earth increases in the presence of the greenhouse gases. Be sure to include reasoning. +he sur"ace o" the earth is in ener!y balance which means that the ener!y or power radiated will e'actly balance the amount o" ener!y or power absorbed% hen !reenhouse !ases are introduced into the atmosphere, the amount o" ener!y absorbed by the sur"ace o" the earth increases because the !reenhouse !ases trap some o" the I- radiation emitted by the sur"ace itsel" and redirect it back down toward the sur"ace where it is absorbed by the sur"ace o" the earth% So not the sur"ace o" the earth is absorbin! both sunli!ht radiation and Iradiation% +he temperature o" the sur"ace will increase until the amount o" ener!y emitted as I- radiation by the sur"ace balances the amount o" ener!y absorbed% d) 9s the /un ages, it will cool and the amount of power it produces will decrease. If at some point in the future, this power drops such that the solar power;m>$ at the earth drops by +&7 to +$%& atts;m>$, what will the surface temperature of the (arth be? 9ssume that the fraction of sunlight absorbed b y the (arth's surface remains the same and that the fraction of the I- radiation emitted by the (arth's surface that reaches space remains the same as it is for today see lecture notes"."
-he Barth is in an energ/ balance. -hat is, the earth is emitting the same amount of energ/ to space as it absorbs from the sunlight. So, the power in from the sun must e?ual the power radiated to space from earth. ower in from the sun ower radiated to space from Barth solar power>m" 5 area of sunlight intersected b/ the earth 5 fraction of sunlight absorbed b/ earth bolt7mann8s constant 5 +earth surface temp4 5 surface area of the earth 5 fraction of surface 2= emitted that makes it to space 2n this problem the solar power per s? meter at the orbit of the Barth has dropped to %"&0 3atts>m ". So solving for the surface temperature we have: +earth surface temp 4 solar power>m" 5 area of sunlight intersected b/ the earth 5 fraction of sunlight absorbed b/ earth bolt7mann8s constant 5 surface area of the earth 5 fraction of surface 2= emitted that makes it to space
+earth surface temp 4 +%"&0 3>m"+ 2 5 ="earth+0.# %.%$ 5 %0%# 3 &.'# 5 %0!( @>+s m" K4 5+ 4 5 2 5 ="earth 5 0.'% %.#' 5 %0# 3>K4 earth surface temp "($ Kelvin +or & degrees colder than toda/ e) If in the future, the amount of greenhouse gases in the atmosphere rise so that +&7 less of the I- radiation emitted from the earth's surface is getting out through the atmosphere than at present, calculate how many degrees elvin hotter the surface of the earth would be? 9ssume that the amount of clouds and other complicated stuff like that does not change."
gain, the Barth is in an energ/ balance. -hat is, the earth is emitting the same amount of energ/ as it absorbs from the sunlight. So, the power in from the sun must e?ual the power radiated to space from earth. ower in from the sun ower radiated to space from Barth solar power>m" 5 area of sunlight intersected b/ the earth 5 fraction of sunlight absorbed b/ earth bolt7mann8s constant 5 +earth surface temp4 5 surface area of the earth 5 fraction of surface 2= emitted that makes it to space 2n this problem, the %01 less of the 2= radiation makes it to space than the present. solving again for the surface temperature we have: +earth surface temp 4 solar power>m" 5 area of sunlight intersected b/ the earth 5 fraction of sunlight absorbed b/ earth bolt7mann8s constant 5 surface area of the earth 5 fraction of surface 2= emitted that makes it to space fraction of surface 2= emitted to space drops from 0.'% to 0.&& ... it is %01 less. +earth surface temp 4 +%$(0 3>m"+ 2 5 ="earth+0.# %."" 5 %0%# 3 &.'# 5 %0!( @>+s m" K4 5+ 4 5 2 5 ="earth 5 +0.&& %.&A 5 %0# 3>K4 earth surface temp "A' Kelvin +or ( degrees warmer than toda/ nother acceptable wa/ to look at this is if the fraction of 2= emitted that makes it to space decreases b/ %01, the 2= emitted b/ the surface must increase b/ about %01 to keep the amount emitted
to space in balance with the power absorbed b/ the earth +actuall/ it8s %%1 but %01 is reasonable for this answer. 2n order for the 2= emitted b/ the surface to increase b/ %01 than the temperature must increase. 2= emitted b/ surface bolt7mann8s constant 5 surface area of the earth 5 +surface temp4 Since the surface area doesn8t change, then +surface temp new4 %.% +that gives new 2= emitted of %01 more +surface temp present 4 surface temp new fourth root + %.%. 5 +surface temp present4 fourth root + %.%. 5 +"(( 4 "A& Kelvin hat is this increase in degrees ahrenheit?
"A& Kelvin "A& ! "#$.%& Celcius "%.(& Celcius -f +A>&-c*$" -empF)ahrenheit +A>& "%.(& * $" #%.$$ )ahrenheit ") In class, we used the temperature sensor to measure the temperature of a hot piece of metal and then saw the effect of sticking a piece of glass in front of the metal. i) If we look at (arth from space with temperature probe similar to the one we used in class and measure the amount of I- radiation emitted to space by the (arth, what temperature would we measure?
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-he temperature probe onl/ measures the amount of 2= that makes it to space. 2n order to balance the power in from the sunlight, the earth must radiate as much power to space as an obect at "&& K. -he surface temperature is onl/ higher than "&& K because not all of the 2= emitted from the surface makes it to space. -he temperature probe cannot tell that the surface is hotter than "&& K, it onl/ knows how much radiation it measures and it measures the same power as is emitted b/ an obect at "&& K. ii) hen the concentration of greenhouse gases increases, most of the I- radiation that escapes to space last interacted with a greenhouse gas molecule at a ...
higher altitude 3hen the atmosphere contains greenhouse gases, an 2= radiation lower altitude
emitted from the surface is absorbed and redirected man/ times before it actuall/ escapes to space. 2f the concentration increases, on average the 2= radiation will have its last interaction with a greenhouse gas at a higher altitude. 3hen there are no greenhouse gases, the 2= radiation that is emitted to space comes directl/ from the ground. s the greenhouse gases increase, the 2= radiation is intercepted at higher altitudes and then redirected.
