Question:
The power radiated by a black body is $P$ and it radiates maximum energy at wavelength, $\lambda_0$. If the temperature of the black body is now changed so that it radiates maximum energy at wavelength $\frac{3}{4} \lambda_0$ , the power radiated by it becomes $nP$. The value of $n$ is
The power radiated by a black body is $P$ and it radiates maximum energy at wavelength, $\lambda_0$. If the temperature of the black body is now changed so that it radiates maximum energy at wavelength $\frac{3}{4} \lambda_0$ , the power radiated by it becomes $nP$. The value of $n$ is
Updated On: Jul 13, 2024
- $\frac{81}{256}$
- $\frac{3}{4}$
- $\frac{256}{81}$
- $\frac{4}{3}$
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The Correct Option is C
Solution and Explanation
We know,
$\lambda_{\max} $ T = constant (Wien's law)
So, $\lambda_{\max_1} T_{1} = \lambda_{\max_2 } T_{2}$
$ \Rightarrow \lambda_{0} T = \frac{3\lambda_{0}}{4} T' $
$\Rightarrow T' = \frac{4}{3} T$
So, $ \frac{P_{2}}{P_{1}} = \left(\frac{T'}{T}\right)^{4} = \left(\frac{4}{3}\right)^{4} = \frac{256}{81} $
$\lambda_{\max} $ T = constant (Wien's law)
So, $\lambda_{\max_1} T_{1} = \lambda_{\max_2 } T_{2}$
$ \Rightarrow \lambda_{0} T = \frac{3\lambda_{0}}{4} T' $
$\Rightarrow T' = \frac{4}{3} T$
So, $ \frac{P_{2}}{P_{1}} = \left(\frac{T'}{T}\right)^{4} = \left(\frac{4}{3}\right)^{4} = \frac{256}{81} $
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Concepts Used:
Photoelectric Effect
When light shines on a metal, electrons can be ejected from the surface of the metal in a phenomenon known as the photoelectric effect. This process is also often referred to as photoemission, and the electrons that are ejected from the metal are called photoelectrons.
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According to Einstein’s explanation of the photoelectric effect :
The energy of photon = energy needed to remove an electron + kinetic energy of the emitted electron
i.e. hν = W + E
Where,
- h is Planck’s constant.
- ν is the frequency of the incident photon.
- W is a work function.
- E is the maximum kinetic energy of ejected electrons: 1/2 mv².
Laws of Photoelectric Effect:
- The photoelectric current is in direct proportion to the intensity of light, for a light of any given frequency; (γ > γ Th).
- There exists a certain minimum (energy) frequency for a given material, called threshold frequency, below which the discharge of photoelectrons stops completely, irrespective of how high the intensity of incident light is.
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- The process of photo-emission is an instantaneous process.