A black body radiates at two temperatures T1 and T₂ such that T_{1 2}, the frequency corresponding to maximum intensity is

(a) less at T₁

(b) more at T₁

(c) equal in the two cases

Please explain the solution in detail.

To tackle the question about the radiation of a black body at two different temperatures, T1 and T2, we can refer to Wien's Displacement Law. This law provides a relationship between the temperature of a black body and the wavelength (or frequency) at which it emits radiation most intensely. Let's break this down step by step.

Wien's Displacement Law

Wien's Displacement Law states that the wavelength of the peak emission (λ_max) of a black body is inversely proportional to its absolute temperature (T). Mathematically, it can be expressed as:

λ_max = b / T

Here, b is Wien's displacement constant, approximately equal to 2898 μm·K. This means that as the temperature increases, the peak wavelength of emitted radiation decreases.

Understanding Frequency and Wavelength

Frequency (ν) and wavelength (λ) are related through the speed of light (c) by the equation:

c = ν × λ

From this relationship, we can see that if the wavelength decreases (as it does with increasing temperature), the frequency must increase. Thus, when we talk about the frequency corresponding to maximum intensity, we can derive that:

ν_max = c / λ_max

Applying the Law to the Given Temperatures

Now, let’s apply this understanding to the two temperatures, T1 and T2, where we assume T2 > T1. According to Wien's Displacement Law:

• For temperature T1, the peak wavelength is λ_max1 = b / T1.
• For temperature T2, the peak wavelength is λ_max2 = b / T2.

Since T2 is greater than T1, it follows that:

• λ_max2 < λ_max1 (the peak wavelength at T2 is shorter).

Consequently, since frequency is inversely related to wavelength, we find:

• ν_max2 > ν_max1 (the peak frequency at T2 is higher).

Conclusion on the Options

Now, looking back at the options provided in the question:

• (a) less at T1
• (b) more at T1
• (c) equal in the two cases

Since we established that the frequency corresponding to maximum intensity increases with temperature, the correct answer is:

(b) more at T1

Final Thoughts

This relationship is fundamental in understanding thermal radiation and has implications in various fields, including astrophysics and engineering. By grasping these concepts, you can better appreciate how temperature influences the emission of radiation from objects, which is crucial in many scientific applications.