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### Energy distribution spectrum of black body radiation

Description→ The energy distribution among the different wavelengths in the spectrum of black body radiation was studied by Lummer and Pringsheim in 1899. There are the following important observations of the study.
1. The energy distribution in the radiation spectrum of the black body is not uniform. As the temperature of the body rises the intensity of radiation for each wavelength increases.

2. At a given temperature, the intensity of radiation increases with increases in wavelength and becomes maximum at a particular wavelength with further in increases wavelength the intensity of radiation decreases.

3.  Energy distribution in the spectrum of black body radiation
4. The points of maximum energy shift towards the shorter wavelength as the temperature increases i.e. $\lambda _{m} \times T=constant$. It is also known as Wein’s displacement law of energy distribution.

5. For a given temperature the total energy of radiation is represented by the area between the curve and the horizontal axis and the area increases with increases of temperature, being directly proportional to the fourth power of absolute temperatures.
The total amount of heat radiated by a perfectly black body per unit area per unit time is directly proportional to the fourth power of its absolute temperature $(T)$.

 $E = \sigma T^{4}$

Where $\sigma$ = Stefan constant having value $\left ({5}\cdot{67}\times{10}^{-8}Wm^{-2}K^{4}\right )$

This is called Stefan-Boltzmann's law of energy distribution.

### Numerical Aperture and Acceptance Angle of the Optical Fibre

Angle of Acceptance → If incident angle of light on the core for which the incident angle on the core-cladding interface equals the critical angle then incident angle of light on the core is called the "Angle of Acceptance. Transmission of light when the incident angle is equal to the acceptance angle If the incident angle is greater than the acceptance angle i.e. $\theta_{i}>\theta_{0}$ then the angle of incidence on the core-cladding interface will be less than the critical angle due to which part of the incident light is transmitted into cladding as shown in the figure below Transmission of light when the incident angle is greater than the acceptance angle If the incident angle is less than the acceptance angle i.e. $\theta_{i}<\theta_{0}$ then the angle of incidence on the core-cladding interface will be greater than the critical angle for which total internal reflection takes place inside the core. As shown in the figure below Transmission of lig

### Fraunhofer diffraction due to a single slit

Let $S$ be a point monochromatic source of light of wavelength $\lambda$ placed at the focus of collimating lens $L_{1}$. The light beam is incident normally from $S$ on a narrow slit $AB$ of width $e$ and is diffracted from it. The diffracted beam is focused at the screen $XY$ by another converging lens $L_{2}$. The diffraction pattern having a central bright band followed by an alternative dark and bright band of decreasing intensity on both sides is obtained. Analytical Explanation: The light from the source $S$ is incident as a plane wavefront on the slit $AB$. According to Huygens's wave theory, every point in $AB$ sends out secondary waves in all directions. The undeviated ray from $AB$ is focused at $C$ on the screen by the lens $L_{2}$ while the rays diffracted through an angle $\theta$ are focussed at point $p$ on the screen. The rays from the ends $A$ and $B$ reach $C$ in the same phase and hence the intensity is maximum. Fraunhofer diffraction due to

### Electromagnetic wave equation in free space

Maxwell's Equations: Maxwell's equation of the electromagnetic wave is a collection of four equations i.e. Gauss's law of electrostatic, Gauss's law of magnetism, Faraday's law of electromotive force, and Ampere's Circuital law. Maxwell converted the integral form of these equations into the differential form of the equations. The differential form of these equations is known as Maxwell's equations. $\overrightarrow{\nabla}. \overrightarrow{E}= \frac{\rho}{\epsilon_{0}}$ $\overrightarrow{\nabla}. \overrightarrow{B}=0$ $\overrightarrow{\nabla} \times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$ $\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \overrightarrow{J}$ Modified Form: $\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \left(\overrightarrow{J}+ \epsilon \frac{ \partial \overrightarrow{E}}{\partial t} \right)$ For free space