Question

show that wien's displacement law can be derived from planck's radiation law​

Answer 1

Derive Wien's displacement law from Planck's law. Proceed as follows:

ρ(ν,T)=2hν3c3(ehνkBT−1)(1)

We need to evaluate the derivative of Equation 1 with respect to ν and set it equal to zero to find the peak wavelength.

ddν{ρ(ν,T)}=ddν⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪2hν3c3(ehνkBT−1)⎫⎭⎬⎪⎪⎪⎪⎪⎪⎪⎪=0(2)

This can be solved via the quotient rule or product rule for differentiation. Selecting the latter for convenience requires rewriting Equation 2 as a product:

ddν{ρ(ν,T)}=2hc3ddν{(ν3)(ehνkBT−1)−1}=0(3)

applying the product rule (and power rule and chain rule)

=2hc3[(3ν2)(ehνkBT−1)−1−(ν3)(ehνkBT−1)−2(hkBT)ehνkBT]=0(4)

so this expression is zero when

(3ν2)(ehνkBT−1)−1=(ν3)(ehνkBT−1)−2(hkBT)ehνkBT(5)

or when simplified

3(ehνkBT−1)−(hvkBT)ehνkBT=0(6)

We can do a substitution u=hνkBT and Equation 6 becomes

3(eu−1)−ueu=0(7)

Finding the solutions to this equation requires using Lambert's W-functions and results numerically in

u=3+W(−3e−3)≈2.8214(8)

so unsubstituting the u variable

u=hνkBT≈2.8214(9)

or

ν≈2.8214kBhT≈(2.8214)(1.38×10−23J/K)6.63×10−34JsT≈(5.8×1010Hz/K)T(10)(11)(12)

The consequence is that the shape of the blackbody radiation function would shift proportionally in frequency with temperature. When Max Planck later formulated the correct blackbody radiation function it did not include Wien's constant explicitly. Rather, Planck's constant h was created and introduced into his new formula. From Planck's constant h and the Boltzmann constant k, Wien's constant (Equation 9 ) can be obtained.