Question

A black body emits radiation of maximum intensity at wavelength '$\lambda$' at temperature $T$ K. Its corresponding wavelength at temperature $1.5 T$ K will be ______.

• A. $\frac{2\lambda}{3}$
• B. $\frac{4\lambda}{3}$
• C. $\frac{16\lambda}{81}$
• D. $\frac{81\lambda}{16}$

Hint:

Always ensure temperatures are in absolute Kelvin scale before applying Wien's Law. Inverse proportionality means if the temperature goes up by a factor of 1.5 ($3/2$), the wavelength must go down by that same factor (multiply by $2/3$).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem relates the peak emission wavelength of a black body to its absolute temperature.

Step 2: Key Formula or Approach:
According to Wien's Displacement Law, the wavelength corresponding to maximum emission intensity ($\lambda_{max}$) is inversely proportional to the absolute temperature ($T$) of the black body.
$$\lambda_{max} T = b \quad \text{(Wien's Constant)}$$
Therefore, $\lambda_1 T_1 = \lambda_2 T_2$.

Step 3: Detailed Explanation:
Given initial state: $\lambda_1 = \lambda$, $T_1 = T$.
Given final state: $\lambda_2 = ?$, $T_2 = 1.5 T = \frac{3}{2} T$.
Apply the law:
$$\lambda \cdot T = \lambda_2 \cdot \left( \frac{3}{2} T \right)$$
Cancel $T$ from both sides:
$$\lambda = \frac{3}{2} \lambda_2$$
Isolate $\lambda_2$:
$$\lambda_2 = \frac{2}{3} \lambda$$

Step 4: Final Answer:
The corresponding wavelength is $\frac{2\lambda}{3}$, matching option (a).