8.13 Use the formula ${\mathrm{}\lambda }_m$ T = 0.29 cmK 

to obtain the characteristic temperature ranges for different parts of the electromagnetic spectrum. What do the numbers that you obtain tell you?

<p><strong>8.13 </strong>A body at a particular temperature produces a continuous spectrum of wavelengths. In case of a black body, the wavelength corresponding to maximum intensity of radiation is given according to Plank’s law. It can be given by the relation,</p><p>T = 0.29 cmK</p><p><span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi mathvariant="normal"> </mi> <mi>λ</mi> </mrow> </mrow> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi> </mi> <mfrac> <mrow> <mrow> <mn>0.29</mn> </mrow> </mrow> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> </mfrac> </math> </span>cm K, where <span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi mathvariant="normal"> </mi> <mi>λ</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="bold-italic">m</mi> </mrow> </mrow> </msub> </math> </span></p><p>= maximum wavelength and T = temperature</p><p>Thus, the temperature for different wavelengths can be obtained as:</p><p>For <span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi mathvariant="normal"> </mi> <mi>λ</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="bold-italic">m</mi> </mrow> </mrow> </msub> </math> </span>&nbsp;=<span contenteditable="false"> <math> <msup> <mrow> <mrow> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>4</mn> <mi> </mi> </mrow> </mrow> </msup> </math> </span>cm, T = 2900 <span contenteditable="false"> <math> <mo>°</mo> <mi>K</mi> </math> </span></p><p>For <span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi mathvariant="normal"> </mi> <mi>λ</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="bold-italic">m</mi> </mrow> </mrow> </msub> </math> </span>&nbsp;=<span contenteditable="false"> <math> <msup> <mrow> <mrow> <mn>5</mn> <mi> </mi> <mo>×</mo> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>5</mn> <mi> </mi> </mrow> </mrow> </msup> </math> </span>cm, T = 5800 <span contenteditable="false"> <math> <mo>°</mo> <mi>K</mi> </math> </span></p><p>For <span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi mathvariant="normal"> </mi> <mi>λ</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="bold-italic">m</mi> </mrow> </mrow> </msub> </math> </span>&nbsp;=<span contenteditable="false"> <math> <msup> <mrow> <mrow> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>6</mn> <mi> </mi> </mrow> </mrow> </msup> </math> </span>&nbsp;cm, T = 290<span contenteditable="false"> <math> <mo>×</mo> <mi> </mi> <msup> <mrow> <mrow> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </msup> </math> </span><span contenteditable="false"> <math> <mo>°</mo> <mi>K</mi> </math> </span>and so on</p><p>The numbers obtained tell us that temperature ranges are required for obtaining radiations in different parts of an electromagnetic spectrum. As the wavelength decreases, the corresponding temperature increases.</p>

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