Question

A black body has wavelength $\lambda_m$ corresponding to maximum energy at 2000 K. Its wavelength corresponding to maximum energy at 3000 K will be:

• A. $\frac{3}{2} \lambda_m$
• B. $\frac{2}{3} \lambda_m$
• C. $\frac{16}{81} \lambda_m$
• D. $\frac{81}{16} \lambda_m$

Answer 1

The Correct Option is B

To solve this problem, we need to apply Wien's Displacement Law, which states that the wavelength of maximum emission of a black body is inversely proportional to its temperature. The formula can be expressed as:

\lambda_m T = b

where \lambda_m is the peak wavelength, T is the absolute temperature, and b is a constant known as Wien's constant.

According to the problem, at 2000 K, the peak wavelength is \lambda_m. When the temperature is increased to 3000 K, we denote the new wavelength as \lambda_m'.

Using Wien's Law for both situations:

1. Initial condition: \lambda_m \times 2000 = b
2. Final condition: \lambda_m' \times 3000 = b

Since b is a constant, we can equate the two expressions:

\lambda_m \times 2000 = \lambda_m' \times 3000

Solve for \lambda_m':

\lambda_m' = \frac{2000}{3000} \times \lambda_m

\lambda_m' = \frac{2}{3} \lambda_m

Thus, the wavelength corresponding to maximum energy at 3000 K is \frac{2}{3} \lambda_m.

Therefore, the correct answer is:

\frac{2}{3} \lambda_m

This corresponds with the given correct answer. This approach utilizes Wien's Displacement Law and validates the given answer by comparing the proportional relationships of temperature and wavelength of a black body.