The Question is from neet-physics electromagnetic-waves , and it was created by the author Mr. ANIL KUMAR.G

Question

A source emits electromagnetic radiation with a peak wavelength of $\lambda_1$ at temperature $T_1$. If the temperature is increased to $T_2 = 4T_1$, what will be the new peak wavelength $\lambda_2$?

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
	**Primary Concept**: Wien's Displacement Law
	**Related Topics**: Blackbody Radiation, Electromagnetic Spectrum
	**Prerequisites**: Relationship between wavelength and frequency, proportionality
## SOLUTION PATHWAY
**Given Information**:
	$\lambda_1$ is the peak wavelength at temperature $T_1$.
	$T_2 = 4T_1$
**Method & Steps**:
	1. **Apply Wien's Displacement Law:**
		Reasoning: Wien's Law states that the peak wavelength of blackbody radiation is inversely proportional to its absolute temperature.
		Formula/Rule: $\lambda_{max} T = b$, where 'b' is Wien's displacement constant.
	2. **Set up the equations:**
		Working: For the initial state: $\lambda_1 T_1 = b$. For the final state: $\lambda_2 T_2 = b$.
		Key Point: Since both products equal 'b', they are equal to each other.
	3. **Solve for $\lambda_2$:**
		Result: $\lambda_1 T_1 = \lambda_2 T_2$. Substituting $T_2 = 4T_1$, we get $\lambda_1 T_1 = \lambda_2 (4T_1)$.  Thus, $\lambda_2 = \lambda_1/4$.
		Verification:  As temperature increases, peak wavelength decreases, which is consistent with Wien's Law.
## OPTION ANALYSIS
[A] ✓: $\lambda_1/4$: This is the correct answer, derived using Wien's Displacement Law.
[B] :x: $\lambda_1/2$: This would imply a temperature increase by a factor of 2, not 4.
[C] :x: $2\lambda_1$: This would imply a temperature decrease, which contradicts the given information.
[D] :x: $4\lambda_1$: This would imply a large temperature decrease, also contradicting the given information.
## LEARNING AIDS
	**Common Mistakes**:
		1. Incorrect Proportionality: Confusing the direct and inverse relationship between temperature and peak wavelength. Remember, higher temperature means shorter wavelength.
		2. Forgetting Wien's Constant:  While not explicitly used in the calculation here, understanding its significance is crucial for a complete understanding of Wien's Law.
	**Quick Tips**:
		1. Visualize the Blackbody Spectrum:  A mental image of the spectrum shifting to shorter wavelengths with increasing temperature can be helpful.
		2. Remember the Inverse Relationship:  A simple phrase like 'higher temperature, shorter wavelength' can help recall the key concept.

Mr. ANIL KUMAR.G · Answer

The correct Option is "λ₁/4"

Mr. ANIL KUMAR.G · Answer

λ₁/2

Mr. ANIL KUMAR.G · Answer

2λ₁

Mr. ANIL KUMAR.G · Answer

4λ₁

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