33:
1335:
2093:
2652:
3051:
2260:
1107:
967:
421:
as wavelength approaches zero (as frequency tends to infinity). Measurements of the spectral emission of actual black bodies revealed that the emission agreed with
Rayleigh's calculation at low frequencies but diverged at high frequencies, reaching a maximum and then falling with frequency, so the
3222:
2823:
2102:
Depending on the application, the Planck function can be expressed in 3 different forms. The first involves energy emitted per unit time per unit area of emitting surface, per unit solid angle, per spectral unit. In this form, the Planck function and associated
Rayleigh–Jeans limits are given by
450:
for the blackbody radiation curve, which accurately described the data at high frequencies. Planck found Wien's original derivation inadequate and devised his own. Then, after learning that the most recent experimental results disagreed with his predictions for low frequencies, Planck revised his
2426:
3410:
1936:
1941:
607:
445:
Rayleigh published his first derivation of the frequency dependence in June 1900. Planck discovered the curve now known as Planck's law in
October of that year and presented it in December. Planck's original intent was to find a satisfactory derivation of
2493:
2902:
2106:
831:
3056:
2657:
824:
2265:
630:. Planck's law does not suffer from an ultraviolet catastrophe and agrees well with the experimental data, but its full significance (which ultimately led to quantum theory) was only appreciated several years later. Since
384:
The
Rayleigh–Jeans law agrees with experimental results at large wavelengths (low frequencies) but strongly disagrees at short wavelengths (high frequencies). This inconsistency between observations and the predictions of
1330:{\displaystyle B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\text{B}}T}}-1}}\approx {\frac {2h\nu ^{3}}{c^{2}}}\cdot {\frac {k_{\text{B}}T}{h\nu }}={\frac {2\nu ^{2}k_{\mathrm {B} }T}{c^{2}}}.}
726:
1046:
155:
1842:
1837:
380:
1753:
488:
438:. Rayleigh believed that this discrepancy could be resolved by the equipartition theorem failing to be valid for high-frequency vibrations, while Jeans argued that the underlying cause was matter and
1404:
2897:
1573:
1466:
735:
2488:
1102:
430:
also presented a derivation of the wavelength dependence in 1903. More complete derivations, which included the proportionality constant, were presented in 1905 by
Rayleigh and Sir
2088:{\displaystyle B_{\lambda }(T)={\frac {2k_{\text{B}}T\left({\frac {c}{\lambda }}\right)^{2}}{c^{2}}}\times {\frac {c}{\lambda ^{2}}}={\frac {2ck_{\text{B}}T}{\lambda ^{4}}}.}
1643:
1607:
241:
182:
1683:
2647:{\displaystyle I(\lambda ,T)={\frac {2\pi hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda k_{\text{B}}T}}-1}}\approx {\frac {2\pi ck_{\text{B}}T}{\lambda ^{4}}}}
1489:
1512:
3046:{\displaystyle u(\lambda ,T)={\frac {8\pi hc}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda k_{\text{B}}T}}-1}}\approx {\frac {8\pi k_{\text{B}}T}{\lambda ^{4}}}}
296:
2255:{\displaystyle B_{\lambda }(T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda k_{\text{B}}T}}-1}}\approx {\frac {2ck_{\text{B}}T}{\lambda ^{4}}}}
633:
265:
210:
962:{\displaystyle {\frac {1}{e^{\frac {hc}{\lambda k_{\text{B}}T}}-1}}\approx {\frac {1}{\frac {hc}{\lambda k_{\text{B}}T}}}={\frac {\lambda k_{\text{B}}T}{hc}}.}
974:
83:
3217:{\displaystyle u(\nu ,T)={\frac {8\pi h\nu ^{3}}{c^{3}}}{\frac {1}{e^{\frac {h\nu }{k_{\text{B}}T}}-1}}\approx {\frac {8\pi k_{\text{B}}T\nu ^{2}}{c^{3}}}.}
2818:{\displaystyle I(\nu ,T)={\frac {2\pi h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\text{B}}T}}-1}}\approx {\frac {2\pi k_{\text{B}}T\nu ^{2}}{c^{2}}}.}
1761:
301:
728:
then in the limit of high temperatures or long wavelengths, the term in the exponential becomes small, and the exponential is well approximated with the
2421:{\displaystyle B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\text{B}}T}}-1}}\approx {\frac {2k_{\text{B}}T\nu ^{2}}{c^{2}}}.}
1692:
3435:
3430:
1350:
2490:
for emitted power integrated over all solid angles. In this form, the Planck function and associated
Rayleigh–Jeans limits are given by
1517:
1409:
3310:
3285:
2899:
for energy per unit volume (energy density). In this form, the Planck function and associated
Rayleigh–Jeans limits are given by
1755:
where both sides now have units of power (energy emitted per unit time) per unit area of emitting surface, per unit solid angle.
422:
total energy emitted is finite. Rayleigh recognized the unphysical behavior of his formula at high frequencies and introduced an
1931:{\displaystyle {\frac {d\nu }{d\lambda }}={\frac {d}{d\lambda }}\left({\frac {c}{\lambda }}\right)=-{\frac {c}{\lambda ^{2}}}.}
1347:
When comparing the frequency- and wavelength-dependent expressions of the
Rayleigh–Jeans law, it is important to remember that
406:
2830:
3247:
602:{\displaystyle B_{\lambda }(T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\frac {hc}{\lambda k_{\text{B}}T}}-1}},}
3237:
65:
3232:
2433:
397:, which gives the correct radiation at all frequencies, has the Rayleigh–Jeans law as its low-frequency limit.
32:
1068:
390:
414:
3346:
1612:
1578:
464:
1685:
has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle,
1645:
has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle,
819:{\displaystyle e^{\frac {hc}{\lambda k_{\text{B}}T}}\approx 1+{\frac {hc}{\lambda k_{\text{B}}T}}.}
439:
219:
160:
3242:
1652:
627:
447:
244:
37:
3415:
1471:
3362:
3306:
3281:
386:
185:
61:
3277:
3271:
413:
dependence of the
Rayleigh–Jeans law based on classical physical arguments, relying upon the
3389:
3354:
1051:
The same argument can be applied to the blackbody radiation expressed in terms of frequency
1494:
614:
435:
427:
426:
cutoff to correct it, but experimentalists found that his cutoff did not agree with data.
281:
482:
394:
41:
3350:
250:
213:
195:
3424:
3393:
3334:
729:
1339:
This last expression is the
Rayleigh–Jeans law in the limit of small frequencies.
17:
431:
268:
45:
3358:
460:
73:
69:
3366:
276:
189:
721:{\displaystyle e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots ,}
1048:
which is identical to the classically derived Rayleigh–Jeans expression.
418:
1041:{\displaystyle B_{\lambda }(T)={\frac {2ck_{\text{B}}T}{\lambda ^{4}}},}
150:{\displaystyle B_{\lambda }(T)={\frac {2ck_{\text{B}}T}{\lambda ^{4}}},}
53:
1832:{\displaystyle B_{\lambda }(T)=B_{\nu }(T){\frac {d\nu }{d\lambda }},}
1468:
Note that these two expressions then have different units, as a step
272:
1758:
Starting with the Rayleigh–Jeans law in terms of wavelength, we get
375:{\displaystyle B_{\nu }(T)={\frac {2\nu ^{2}k_{\text{B}}T}{c^{2}}}.}
76:
at a given temperature through classical arguments. For wavelength
31:
1343:
Consistency of frequency- and wavelength-dependent expressions
3380:
Kragh, H. (2000). "Max Planck: the reluctant revolutionary".
2430:
Alternatively, Planck's law can be written as an expression
417:. This law predicted an energy output that diverges towards
1748:{\displaystyle B_{\lambda }\,d\lambda =dP=B_{\nu }\,d\nu ,}
451:
calculation, obtaining what is now called Planck's law.
1399:{\displaystyle {\frac {dP}{d\lambda }}=B_{\lambda }(T)}
971:
This results in Planck's blackbody formula reducing to
2833:
3059:
2905:
2660:
2496:
2436:
2268:
2109:
1944:
1845:
1764:
1695:
1655:
1615:
1581:
1520:
1497:
1474:
1412:
1353:
1110:
1071:
977:
834:
738:
636:
491:
304:
284:
253:
222:
198:
163:
86:
2892:{\textstyle u(\nu ,T)={\frac {4\pi }{c}}B_{\nu }(T)}
3216:
3045:
2891:
2817:
2646:
2482:
2420:
2254:
2087:
1930:
1831:
1747:
1677:
1637:
1601:
1567:
1506:
1483:
1460:
1398:
1329:
1096:
1040:
961:
818:
720:
601:
374:
290:
259:
235:
204:
176:
149:
27:Approximation of a black body's spectral radiance
1568:{\displaystyle B_{\lambda }(T)\neq B_{\nu }(T),}
1461:{\displaystyle {\frac {dP}{d\nu }}=B_{\nu }(T).}
188:(the power emitted per unit emitting area, per
1689:. To be consistent, we must use the equality
1065:. In the limit of small frequencies, that is
8:
2827:In other cases, Planck's law is written as
1491:in wavelength is not equivalent to a step
3203:
3192:
3179:
3166:
3141:
3126:
3116:
3108:
3097:
3081:
3058:
3035:
3021:
3008:
2983:
2965:
2955:
2947:
2927:
2904:
2874:
2855:
2832:
2804:
2793:
2780:
2767:
2742:
2727:
2717:
2709:
2698:
2682:
2659:
2636:
2622:
2606:
2581:
2563:
2553:
2545:
2534:
2518:
2495:
2483:{\displaystyle I(\nu ,T)=\pi B_{\nu }(T)}
2465:
2435:
2407:
2396:
2383:
2373:
2348:
2333:
2323:
2315:
2304:
2291:
2273:
2267:
2244:
2230:
2217:
2192:
2174:
2164:
2156:
2145:
2132:
2114:
2108:
2074:
2060:
2047:
2036:
2027:
2016:
2005:
1991:
1977:
1967:
1949:
1943:
1917:
1908:
1888:
1869:
1846:
1844:
1806:
1791:
1769:
1763:
1735:
1729:
1706:
1700:
1694:
1660:
1654:
1620:
1614:
1591:
1580:
1547:
1525:
1519:
1496:
1473:
1440:
1413:
1411:
1381:
1354:
1352:
1316:
1301:
1300:
1290:
1280:
1257:
1250:
1239:
1228:
1215:
1190:
1175:
1165:
1157:
1146:
1133:
1115:
1109:
1085:
1070:
1027:
1013:
1000:
982:
976:
936:
926:
910:
888:
863:
845:
835:
833:
801:
783:
761:
743:
737:
693:
687:
668:
662:
641:
635:
574:
556:
546:
538:
527:
514:
496:
490:
361:
347:
337:
327:
309:
303:
283:
252:
227:
221:
197:
168:
162:
136:
122:
109:
91:
85:
3259:
463:empirically obtained an expression for
3276:. Cambridge University Press. p.
3265:
3263:
1097:{\displaystyle h\nu \ll k_{\text{B}}T}
36:Comparison of Rayleigh–Jeans law with
7:
3416:Derivation of modes a wave in a cube
3328:
3326:
3324:
3322:
3303:Radiative Processes in Astrophysics
1575:even after substituting the value
1302:
442:not being in thermal equilibrium.
25:
3335:"Einstein and the quantum theory"
3273:Astronomy: A Physical Perspective
2098:Other forms of Rayleigh–Jeans law
467:expressed in terms of wavelength
3411:Derivation of Rayleigh–Jeans law
1638:{\displaystyle B_{\lambda }(T)}
1602:{\displaystyle \lambda =c/\nu }
405:In 1900, the British physicist
3075:
3063:
2921:
2909:
2886:
2880:
2849:
2837:
2676:
2664:
2512:
2500:
2477:
2471:
2452:
2440:
2285:
2279:
2126:
2120:
1961:
1955:
1803:
1797:
1781:
1775:
1672:
1666:
1632:
1626:
1559:
1553:
1537:
1531:
1452:
1446:
1393:
1387:
1127:
1121:
994:
988:
508:
502:
321:
315:
103:
97:
1:
3436:Obsolete theories in physics
3431:Foundational quantum physics
298:, the expression is instead
236:{\displaystyle k_{\text{B}}}
177:{\displaystyle B_{\lambda }}
44:, for a body of 5800 K
1678:{\displaystyle B_{\nu }(T)}
60:is an approximation to the
3452:
3394:10.1088/2058-7058/13/12/34
3301:Rybicki; Lightman (2004).
1514:in frequency. Therefore,
455:Comparison to Planck's law
3359:10.1103/RevModPhys.51.863
3339:Reviews of Modern Physics
3305:. Wiley. pp. 20–28.
1484:{\displaystyle d\lambda }
389:is commonly known as the
66:electromagnetic radiation
3270:Kutner, Mark L. (2003).
192:, per unit wavelength),
3333:Pais, A. (1979-10-01).
3248:Sakuma–Hattori equation
3238:Wien's displacement law
391:ultraviolet catastrophe
3218:
3047:
2893:
2819:
2648:
2484:
2422:
2256:
2089:
1932:
1833:
1749:
1679:
1639:
1603:
1569:
1508:
1485:
1462:
1400:
1331:
1098:
1042:
963:
820:
722:
603:
401:Historical development
376:
292:
261:
237:
206:
178:
151:
49:
3219:
3048:
2894:
2820:
2649:
2485:
2423:
2257:
2090:
1933:
1834:
1750:
1680:
1640:
1604:
1570:
1509:
1507:{\displaystyle d\nu }
1486:
1463:
1401:
1332:
1099:
1043:
964:
821:
723:
604:
434:and independently by
415:equipartition theorem
377:
293:
262:
238:
207:
179:
152:
35:
3233:Stefan–Boltzmann law
3057:
2903:
2831:
2658:
2494:
2434:
2266:
2107:
1942:
1843:
1762:
1693:
1653:
1613:
1579:
1518:
1495:
1472:
1410:
1351:
1108:
1069:
975:
832:
736:
634:
489:
465:black-body radiation
302:
291:{\displaystyle \nu }
282:
251:
220:
196:
161:
84:
3351:1979RvMP...51..863P
1647:per unit wavelength
730:Taylor polynomial's
454:
440:luminiferous aether
3243:Wien approximation
3214:
3043:
2889:
2815:
2644:
2480:
2418:
2252:
2085:
1928:
1829:
1745:
1687:per unit frequency
1675:
1635:
1599:
1565:
1504:
1481:
1458:
1396:
1327:
1094:
1038:
959:
816:
732:first-order term:
718:
628:Boltzmann constant
599:
372:
288:
257:
245:Boltzmann constant
233:
202:
174:
147:
58:Rayleigh–Jeans law
50:
38:Wien approximation
18:Rayleigh-Jeans law
3209:
3182:
3161:
3151:
3144:
3114:
3041:
3024:
3003:
2993:
2986:
2953:
2868:
2810:
2783:
2762:
2752:
2745:
2715:
2642:
2625:
2601:
2591:
2584:
2551:
2413:
2386:
2368:
2358:
2351:
2321:
2250:
2233:
2212:
2202:
2195:
2162:
2080:
2063:
2042:
2022:
1999:
1980:
1923:
1896:
1882:
1864:
1824:
1431:
1372:
1322:
1275:
1260:
1245:
1210:
1200:
1193:
1163:
1088:
1033:
1016:
954:
939:
921:
920:
913:
883:
873:
866:
811:
804:
771:
764:
707:
682:
594:
584:
577:
544:
448:Wien's expression
387:classical physics
367:
350:
260:{\displaystyle T}
230:
205:{\displaystyle c}
186:spectral radiance
142:
125:
68:as a function of
62:spectral radiance
16:(Redirected from
3443:
3398:
3397:
3377:
3371:
3370:
3330:
3317:
3316:
3298:
3292:
3291:
3267:
3223:
3221:
3220:
3215:
3210:
3208:
3207:
3198:
3197:
3196:
3184:
3183:
3180:
3167:
3162:
3160:
3153:
3152:
3150:
3146:
3145:
3142:
3135:
3127:
3117:
3115:
3113:
3112:
3103:
3102:
3101:
3082:
3052:
3050:
3049:
3044:
3042:
3040:
3039:
3030:
3026:
3025:
3022:
3009:
3004:
3002:
2995:
2994:
2992:
2988:
2987:
2984:
2974:
2966:
2956:
2954:
2952:
2951:
2942:
2928:
2898:
2896:
2895:
2890:
2879:
2878:
2869:
2864:
2856:
2824:
2822:
2821:
2816:
2811:
2809:
2808:
2799:
2798:
2797:
2785:
2784:
2781:
2768:
2763:
2761:
2754:
2753:
2751:
2747:
2746:
2743:
2736:
2728:
2718:
2716:
2714:
2713:
2704:
2703:
2702:
2683:
2653:
2651:
2650:
2645:
2643:
2641:
2640:
2631:
2627:
2626:
2623:
2607:
2602:
2600:
2593:
2592:
2590:
2586:
2585:
2582:
2572:
2564:
2554:
2552:
2550:
2549:
2540:
2539:
2538:
2519:
2489:
2487:
2486:
2481:
2470:
2469:
2427:
2425:
2424:
2419:
2414:
2412:
2411:
2402:
2401:
2400:
2388:
2387:
2384:
2374:
2369:
2367:
2360:
2359:
2357:
2353:
2352:
2349:
2342:
2334:
2324:
2322:
2320:
2319:
2310:
2309:
2308:
2292:
2278:
2277:
2261:
2259:
2258:
2253:
2251:
2249:
2248:
2239:
2235:
2234:
2231:
2218:
2213:
2211:
2204:
2203:
2201:
2197:
2196:
2193:
2183:
2175:
2165:
2163:
2161:
2160:
2151:
2150:
2149:
2133:
2119:
2118:
2094:
2092:
2091:
2086:
2081:
2079:
2078:
2069:
2065:
2064:
2061:
2048:
2043:
2041:
2040:
2028:
2023:
2021:
2020:
2011:
2010:
2009:
2004:
2000:
1992:
1982:
1981:
1978:
1968:
1954:
1953:
1937:
1935:
1934:
1929:
1924:
1922:
1921:
1909:
1901:
1897:
1889:
1883:
1881:
1870:
1865:
1863:
1855:
1847:
1838:
1836:
1835:
1830:
1825:
1823:
1815:
1807:
1796:
1795:
1774:
1773:
1754:
1752:
1751:
1746:
1734:
1733:
1705:
1704:
1684:
1682:
1681:
1676:
1665:
1664:
1644:
1642:
1641:
1636:
1625:
1624:
1608:
1606:
1605:
1600:
1595:
1574:
1572:
1571:
1566:
1552:
1551:
1530:
1529:
1513:
1511:
1510:
1505:
1490:
1488:
1487:
1482:
1467:
1465:
1464:
1459:
1445:
1444:
1432:
1430:
1422:
1414:
1405:
1403:
1402:
1397:
1386:
1385:
1373:
1371:
1363:
1355:
1336:
1334:
1333:
1328:
1323:
1321:
1320:
1311:
1307:
1306:
1305:
1295:
1294:
1281:
1276:
1274:
1266:
1262:
1261:
1258:
1251:
1246:
1244:
1243:
1234:
1233:
1232:
1216:
1211:
1209:
1202:
1201:
1199:
1195:
1194:
1191:
1184:
1176:
1166:
1164:
1162:
1161:
1152:
1151:
1150:
1134:
1120:
1119:
1103:
1101:
1100:
1095:
1090:
1089:
1086:
1064:
1047:
1045:
1044:
1039:
1034:
1032:
1031:
1022:
1018:
1017:
1014:
1001:
987:
986:
968:
966:
965:
960:
955:
953:
945:
941:
940:
937:
927:
922:
919:
915:
914:
911:
901:
893:
889:
884:
882:
875:
874:
872:
868:
867:
864:
854:
846:
836:
825:
823:
822:
817:
812:
810:
806:
805:
802:
792:
784:
773:
772:
770:
766:
765:
762:
752:
744:
727:
725:
724:
719:
708:
706:
698:
697:
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615:Planck constant
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436:Albert Einstein
428:Hendrik Lorentz
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3405:External links
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3345:(4): 863–914.
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3388:(12): 31–36.
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3382:Physics World
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407:Lord Rayleigh
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1686:
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1338:
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970:
827:
619:
610:
483:Planck's law
477:
473:
469:
458:
444:
423:
410:
409:derived the
404:
395:Planck's law
383:
77:
57:
51:
42:Planck's law
29:
432:James Jeans
269:temperature
46:temperature
3425:Categories
3254:References
1649:, whereas
1609:, because
461:Max Planck
74:black body
70:wavelength
3367:0034-6861
3190:ν
3173:π
3164:≈
3155:−
3133:ν
3095:ν
3088:π
3067:ν
3033:λ
3015:π
3006:≈
2997:−
2977:λ
2945:λ
2934:π
2913:λ
2876:ν
2862:π
2841:ν
2791:ν
2774:π
2765:≈
2756:−
2734:ν
2696:ν
2689:π
2668:ν
2634:λ
2613:π
2604:≈
2595:−
2575:λ
2543:λ
2525:π
2504:λ
2467:ν
2459:π
2444:ν
2394:ν
2371:≈
2362:−
2340:ν
2302:ν
2275:ν
2242:λ
2215:≈
2206:−
2186:λ
2154:λ
2116:λ
2072:λ
2034:λ
2025:×
1997:λ
1951:λ
1915:λ
1906:−
1894:λ
1879:λ
1861:λ
1853:ν
1821:λ
1813:ν
1793:ν
1771:λ
1740:ν
1731:ν
1711:λ
1702:λ
1662:ν
1622:λ
1597:ν
1583:λ
1549:ν
1541:≠
1527:λ
1502:ν
1479:λ
1442:ν
1428:ν
1383:λ
1369:λ
1288:ν
1272:ν
1248:⋅
1226:ν
1213:≈
1204:−
1182:ν
1144:ν
1117:ν
1079:≪
1076:ν
1025:λ
984:λ
930:λ
904:λ
886:≈
877:−
857:λ
795:λ
775:≈
755:λ
713:⋯
588:−
568:λ
536:λ
498:λ
335:ν
311:ν
286:ν
277:frequency
190:steradian
170:λ
134:λ
93:λ
3227:See also
459:In 1900
419:infinity
80:, it is
3347:Bibcode
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