Wien's Law and Blackbody Radiation Spectra

1. The Blackbody Radiation Spectrum

When an object radiates heat, it doesn't emit energy at just a single wavelength. Instead, the energy is distributed across a continuous spectrum of wavelengths. For a theoretical ideal emitter, a blackbody, the shape of this energy distribution curve depends only on its temperature.

To visualize this, let's watch a short video that introduces the experimental basis for Wien's analysis.

As you saw, the graph plots the intensity of radiation at each specific wavelength. There are two key features to notice on these curves:

1. The Area Under the Curve: The total area under the curve represents the total power radiated per unit area. As temperature increases, the area grows rapidly. This corresponds to the Stefan-Boltzmann law ($P\propto T^4$) we studied previously.
2. The Peak of the Curve: For any given temperature, there is a specific wavelength, ${\lambda }_{max}$, at which the radiation intensity is at its maximum. As the temperature increases, this peak shifts to shorter wavelengths.

This shift is what Wien's displacement law describes.

2. Wien's Displacement Law

Wien's experiments led to a simple but profound mathematical relationship between the peak wavelength and the absolute temperature of a blackbody.

Let's continue with the same video to get a formal statement of the law.

To summarize the key points:

Wien's Displacement Law states that the wavelength of maximum emission, ${\lambda }_{max}$, is inversely proportional to the absolute temperature $T$ of the blackbody.
${\lambda }_{max}T=b$
where $b$ is Wien's displacement constant, approximately $2.898\times 10^{-3}\text{ m}\cdot \text{K}$.

This simple law elegantly explains the changing color of a heated object. As $T$ increases, ${\lambda }_{max}$ decreases, moving from the infrared, through red, orange, and towards the blue/violet end of the spectrum. The sun, at about 5800 K, has its peak emission in the visible spectrum, which is why it appears white to us.

Wien's 5th Power Law is an extension that tells us the peak intensity on the spectral curve, $I_{\lambda ,max}$, is proportional to the fifth power of the absolute temperature.
$I_{\lambda ,max}\propto T^5$

A Deeper Look: The Origin and a Subtle Pitfall

Given your background, you'll appreciate that these laws aren't just empirical fits. They are direct consequences of Planck's Law of Blackbody Radiation, a cornerstone of quantum mechanics. Planck derived a function $I(\lambda ,T)$ that perfectly describes the entire spectral curve.

Wien's displacement law is derived by finding the maximum of this function, which involves taking the derivative with respect to $\lambda$ and setting it to zero:
$\frac{dI(\lambda ,T)}{d\lambda }=0$
Solving this equation for $\lambda$ yields the relation ${\lambda }_{max}T=\text{constant}$. This places Wien's law on a solid theoretical foundation.

A Conceptual Pitfall:
It's tempting to assume that if you find the frequency $f_{max}$ corresponding to the peak of the frequency spectrum, it would simply be $f_{max}=c/{\lambda }_{max}$. This is incorrect. The reason is subtle: the spectral intensity is a density function. $I_{\lambda }$ is power per unit wavelength, while $I_f$ is power per unit frequency. Because $\lambda$ and $f$ are non-linearly related ($d\lambda =-c/f^{2}df$), a uniform interval in wavelength does not correspond to a uniform interval in frequency. Differentiating Planck's law with respect to frequency yields a different peak. This is a classic trap in advanced conceptual questions.

3. Analyzing Spectra and Solving JEE Advanced Problems

Now, let's apply these concepts, which is the core skill for your exam preparation. The most common use of Wien's law in JEE Advanced is to determine the temperature of a body when its peak emission wavelength is known, often as a first step before applying the Stefan-Boltzmann law.

Let's look at some typical scenarios.

Multi-Step Problem Walkthrough

Let's tackle a typical JEE Advanced problem that combines Wien's law and the Stefan-Boltzmann law.

Problem Statement (Q14): Three discs, A, B, and C having radii 2 m, 4 m, and 6 m respectively are coated with carbon black. The wavelengths corresponding to maximum intensity are 300 nm, 400 nm, and 500 nm, respectively. The power radiated by them are $Q_A$, $Q_B$, and $Q_C$. Which is maximum?

Thought Process & Solution:

1. Deconstruct the Goal: The question asks to compare the total power radiated ($Q_A,Q_B,Q_C$).

2. Identify the Governing Law for Power: The total power radiated is given by the Stefan-Boltzmann law: $Q=ϵ\sigma A{T}^4$. Since they are coated with carbon black, we assume they are ideal blackbodies, so $ϵ=1$. The area $A$ is $\pi r^2$. So, $Q\propto r^2{T}^4$.

3. Identify the Missing Variable: We have the radii ($r$), but we don't have the temperatures ($T$).

4. Find the Connection: The problem gives us the peak wavelengths (${\lambda }_{max}$). This is the key. We can use Wien's displacement law to relate ${\lambda }_{max}$ to $T$.

5. Apply Wien's Law: We know ${\lambda }_{max}T=b$, which means $T=b/{\lambda }_{max}$, or $T\propto 1/{\lambda }_{max}$.

6. Combine the Laws: Now substitute the expression for $T$ into the power equation:
   $Q\propto r^2{T}^4\propto r^2{\left(\frac{1}{{\lambda }_{max}}\right)}^4$
   $Q\propto \frac{r^2}{{\lambda }_{max}^4}$
   This gives us a direct way to compare the powers using the given information.

7. Calculate the Ratios: Now we can find the ratio $Q_A:Q_B:Q_C$. Let's use the values in units of (m) for radius and (100 nm) for wavelength to keep numbers simple.

   • $r_A=2,{\lambda }_A=3$
   • $r_B=4,{\lambda }_B=4$
   • $r_C=6,{\lambda }_C=5$

   $Q_A\propto \frac{2^2}{3^4}=\frac{4}{81}\approx 0.0494$
   $Q_B\propto \frac{4^2}{4^4}=\frac{1}{4^2}=\frac{1}{16}=0.0625$
   $Q_C\propto \frac{6^2}{5^4}=\frac{36}{625}=0.0576$

8. Conclusion: Comparing the values, $Q_B$ is the largest.

This systematic approach—identifying the goal, finding the right physical laws, and combining them algebraically—is crucial for solving complex JEE problems.

Conclusion

Today we've delved into the "color" of thermal radiation, moving beyond just the total energy. You are now equipped to analyze blackbody spectra and solve problems using Wien's displacement law.

Key Takeaways:

• A hot body emits radiation across a spectrum of wavelengths, with a peak intensity at a specific wavelength, ${\lambda }_{max}$.
• Wien's Displacement Law (${\lambda }_{max}T=b$) provides the inverse relationship between this peak wavelength and the body's absolute temperature.
• This law explains why objects change color as they are heated.
• A common problem-solving pattern for JEE Advanced involves using Wien's law to find temperature information from spectral data, and then using that temperature in the Stefan-Boltzmann law to calculate total power.

Preview of the Next Lesson:
We have now studied the three modes of heat transfer—conduction, convection, and radiation—in detail. In our next lesson, we will begin to integrate these concepts. We will learn how to formulate the steady-state energy balance for systems where conduction, convection, and radiation occur simultaneously. This will involve setting up and solving energy balance equations for real-world scenarios, such as a heated object losing heat to the air and its distant surroundings at the same time.