Modern Physics For Scientists And Engineers
Modern Physics For Scientists And Engineers
2nd Edition
ISBN: 9781938787751
Author: Taylor, John R. (john Robert), Zafiratos, Chris D., Dubson, Michael Andrew
Publisher: University Science Books,
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Chapter 4, Problem 4.1P
To determine

The sketch of the graph for the given function and to explain the trends in the graph.

Expert Solution & Answer
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Answer to Problem 4.1P

The graph for the function I(λ,T)=2πhc2λ51ehc/λ k BT1 is shown in figure (1).

Explanation of Solution

Given:

The Planck distribution function for blackbody radiation is given by,

  I(λ,T)=2πhc2λ51ehc/λ k BT1 ...... (I)

Calculation:

Consider the term,

  2πhc2λ5

The value of the term is calculated as,

  2πhc2λ5=2π( 6.63× 10 34 J-s) ( 3× 10 8 m/s )2λ5=3.75× 10 16J- m 2/sλ5

Consider the term,

  hckBT

The value of the term is calculated as,

  hckBT=( 6.63× 10 34 J-s)( 3× 10 8 m/s )( 1.38× 10 23 J/K )( 1500K)=9.6087×106m

Thus, the Planck's distribution function of a blackbody radiation for a fixed temperature T=1500K is given by,

  I(λ,T)=3.75×1016J-m2/sλ51e( 9.6087× 10 6 m)/λ1

The following table shows the Planck's intensity distribution for different wavelengths at a fixed temperature.

Wavelength (λ,m)I(λ)
00
0.000012323463782
0.00002189999621
0.000055663546.65
0.0001371821.542
0.000223810.7114

The following figure shows the Planck's intensity distribution for different wavelengths at a fixed temperature.

  Modern Physics For Scientists And Engineers, Chapter 4, Problem 4.1P

Figure (1)

From the graph, it can be concluded that the intensity of blackbody radiation for a fixed temperature, is maximum at a particular wavelength and it becomes zero with the increase in wavelength. With the increase in temperature, the maximum intensity of a radiation shift towards the shorter wavelengths.

Conclusion:

Therefore, the graph for the function I(λ,T)=2πhc2λ51ehc/λ k BT1 is shown in figure (1).

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