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Showing posts from February, 2023

### Eigen value of the momentum of a particle in one dimension box or infinite potential well

Equation of eigen value of the momentum of a particle in one dimension box: The eigen value of the momentum $P_{n}$ of a particle in one dimension box moving along the x-axis is given by $P^{2}_{n} = 2 m E_{n}$ $P^{2}_{n} = 2 m \frac{n^{2} \pi^{2} \hbar^{2}}{2 m L^{2}} \qquad \left( \because E_{n}= \frac{n^{2} \pi^{2} \hbar^{2}}{2 m L^{2}} \right)$ $P^{2}_{n} = \frac{n^{2} \pi^{2} \hbar^{2}}{L^{2}}$ $P_{n} = \pm \frac{n \pi \hbar}{L}$ $P_{n} = \pm \frac{n h}{2L} \qquad \left( \hbar = \frac{h}{2 \pi} \right)$ The $\pm$ sign indicates that the particle is moving back and forth in the infinite potential box. The above equation shows that eigen value of the momentum of the particle is discrete and the difference between the momentum corresponding to two consecutive energy levels is always constant and equal to $\frac{h}{2L}$

### Electric field intensity due to uniformly charged solid sphere (Conducting and Non-conducting)

A.) Electric field intensity at different points in the field due to the uniformly charged solid conducting sphere: Let us consider, A solid conducting sphere that has a radius $R$ and charge $+q$ is distributed on the surface of the sphere in a uniform manner. Now find the electric field intensity at different points due to the solid-charged conducting sphere. These different points are: Electric field intensity outside the solid conducting sphere Electric field intensity on the surface of the solid conducting sphere Electric field intensity inside the solid conducting sphere 1.) Electric field intensity outside the solid conducting sphere: If $O$ is the center of solid conducting spherical then the electric field intensity outside of the sphere can be determined by the following steps → First, take the point $P$ outside the sphere Draw a spherical surface of radius r which passes through point $P$. This hypothetical surface is known

### The electric potential at different points (like on the axis, equatorial, and at any other point) of the electric dipole

Electric Potential due to an Electric Dipole: The electric potential due to an electric dipole can be measured at different points: The electric potential on the axis of the electric dipole The electric potential on the equatorial line of the electric dipole The electric potential at any point of the electric dipole 1. The electric potential on the axis of the electric dipole: Let us consider, An electric dipole AB made up of two charges of -q and +q coulomb is placed in a vacuum or air at a very small distance of $2l$. Let a point $P$ is on the axis of an electric dipole and place at a distance $r$ from the center point $O$ of the electric dipole. Now put the test charged particle $q_{0}$ at point $P$ for the measurement of electric potential due to dipole's charges. So Electric potential at point $P$ due $+q$ charge of electric dipole→ $V_{+q}=\frac{1}{4\pi \epsilon_{0}} \frac{q}{r-l}$ The electric potential at point $P$ due $-q$ charge of elect

### Electromagnetic Wave Equation in Conducting Media (i.e. Lossy dielectric or Partially Conducting)

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: $\overrightarrow{\nabla} \times \overrightarrow{B}= \mu \overrightarrow{J}+\mu \epsilon \frac{\partial \overrightarrow{E}}{\partial t}$ For Conducting Media: Cur