18nucleons

# FRAUNHOFER DIFFRACTION BY IDENTICAL APERTURES From Two Slits to Diffraction Grating | Detailed Derivation 1. Diffraction by N Identical Apertures 1.1 Starting from the Diffraction Integral From the general Fraunhofer diffraction integral, the field at a point \( P \) is: \[ u(P) = C \iint P(\xi,\eta) e^{ik(l\xi + m\eta)} d\xi d\eta \] where \( l = \sin\theta_x \) and \( m = \sin\theta_y \). This represents the Fourier transform of the aperture function. 1.2 N Apertures Summation Let there be \( N \) identical apertures centered at \( O_1, O_2, \dots O_N \). The total field is the sum of integrals over each aperture: \[ u(P) = C \sum_{n=1}^{N} \iint_{S_n} e^{ik(l\xi + m\eta)} d\xi d\eta \] By shifting coordinates such that \( \xi = \xi' + \xi_n \) and \( \eta = \eta' + \eta_n \), we can separate the integral into a sing...

RESOLVING POWER & DIFFRACTION LIMITS From Circular Aperture to Microscope Resolution 1. Diffraction by a Circular Aperture (Foundation) For a circular aperture of radius \( a \), the Fraunhofer diffraction intensity is given by: \[ I(\theta) = I_0 \left[ \frac{2J_1(v)}{v} \right]^2 \] where the dimensionless parameter \( v \) is: \[ v = ka \sin \theta = \frac{2\pi}{\lambda} a \sin \theta \] Here, \( J_1(v) \) is the first-order Bessel function of the first kind. Condition for the First Minimum The first diffraction minimum (dark ring) occurs when: \[ J_1(v) = 0 \] The first zero of the Bessel function is mathematically determined as: \[ v = 3.832 \] Substituting this back into the expression for \( v \): \[ k a \sin \theta = 3.832 \] \[ \frac{2\pi}{\lamb...

# Derivation of Resolving Power of a Telescope Let us derive the expression for the limit of resolution (\( d\theta \)) and the resolving power of a telescope objectively using the principles of Fraunhofer diffraction. 1. The Setup Consider a telescope with an objective lens of diameter (aperture) \( D \). Imagine two distant stars, \( S_1 \) and \( S_2 \), located very close to each other in the sky. The light rays from the first star \( S_1 \) arrive parallel to the principal axis and form a diffraction pattern centered at \( P_1 \). The light rays from the second star \( S_2 \) arrive at a small angle \( d\theta \) relative to \( S_1 \) and form a diffraction pattern centered at \( P_2 \). 2. Rayleigh's Criterion for Resolution According to Rayleigh's criterion, the two stars are just resolved when the central maximum of the diffraction pattern of \( S_2 \) falls exactly on the first minimum (dark ri...

RESOLVING POWER OF A TELESCOPE Optics & Optical Instruments | Detailed Derivation 1. Introduction and Definitions The primary function of a telescope is to gather light from distant objects and form a bright image. However, a critical measure of its quality is its ability to distinguish two closely spaced distant objects (like a double star system) as separate entities. This ability is called the Resolving Power . Limit of Resolution (\( d\theta \)): The minimum angular separation between two distant point objects such that they are just resolved (seen as separate) by the telescope. Resolving Power (R.P.): It is defined as the reciprocal of the limit of resolution. It quantifies the ability of the telescope to distinguish fine angular details. \[ \text{Resolving Power} = \frac{1}{\text{Limit of Resolution}} = \frac{1}{d\theta...

RESOLVING POWER OF A MICROSCOPE Optics & Optical Instruments | Detailed Derivation 1. Introduction and Definitions In optical instruments, the primary function is often thought to be magnification. However, mere magnification is useless if the image lacks detail. The ability of an instrument to show two closely spaced objects as separate and distinct is called its Resolving Power . Limit of Resolution \( d \): The minimum linear distance between two point objects such that they are just resolved (seen as separate) by the optical system. Resolving Power (R.P.): It is defined as the reciprocal of the limit of resolution. It quantifies the ability of the microscope to distinguish fine details. \[ \text{Resolving Power} = \frac{1}{\text{Limit of Resolution}} = \frac{1}{d} \] 2. Principle: Rayleigh's...

Rectangular Waveguide: Complete Theory from First Principles We consider a rectangular metallic waveguide of cross-section \[ 0 \le x \le a, \quad 0 \le y \le b \] with propagation along the z-direction. The walls are assumed to be perfect electric conductors (PEC) and the interior is vacuum (\(\varepsilon_0, \mu_0\)). 1. Governing Equations in the Waveguide Inside the waveguide there are no free charges and no currents. Maxwell’s equations reduce to: \[ \nabla \times \mathbf{E} = -\mu_0 \frac{\partial \mathbf{H}}{\partial t} \] \[ \nabla \times \mathbf{H} = \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \] Assume monochromatic time dependence: \[ \mathbf{E}, \mathbf{H} \propto e^{-j\omega t} \] Then: \[ \nabla \times \mathbf{E} = -j\omega \mu_0 \mathbf{H} \] \[ \nabla \times \mathbf{H} = j\omega \epsilon_0 \mathbf{E} \] Each field component satisfies the Helmholtz equation: \[ \left(\nab...

Electromagnetic Waves in a Conducting Medium: The Complete Theory This document details the propagation of electromagnetic waves in a material that has conductivity \(\sigma \neq 0\). Unlike in a vacuum, the wave loses energy to the medium (attenuation) and the electric and magnetic fields oscillate out of phase. 1. The Medium Parameters and Maxwell's Equations We consider a medium characterized by: Permittivity (\(\varepsilon\)): Governs the electric displacement field \(\vec{D} = \varepsilon \vec{E}\). Permeability (\(\mu\)): Governs the magnetic flux density \(\vec{B} = \mu \vec{H}\). Conductivity (\(\sigma\)): Governs the conduction current density \(\vec{J} = \sigma \vec{E}\) (Ohm's Law). 1.1 Maxwell's Equations for Conducting Media In a linear, isotropic, homogeneous conductor, we start with the general Maxwell equations: \(\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon}\) (Gauss's Law) \(\nabla \...

Plasma is often described as the "fourth state of matter." Physically, it is an ionized gas consisting of free electrons and ions. Unlike a neutral gas, a plasma is dominated by electromagnetic forces. To understand how electromagnetic waves (like light or radio waves) travel through this medium, we must return to the fundamental microscopic form of Maxwell's Equations. [Image of electromagnetic wave propagation in plasma] 1. Maxwell’s Equations (Microscopic Form) In a dielectric solid, we often use macroscopic fields (\(\mathbf{D}\) and \(\mathbf{H}\)) to hide the complex atomic behavior. In plasma physics, because the charges are free and mobile, it is clearer to use the vacuum equations and treat the charge density \(\rho\) and current density \(\mathbf{J}\) explicitly. The governing equations are: Gauss's Law: \[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \] Gauss's Law for Magnetism: \[ \nabl...

When electromagnetic waves travel through matter rather than a vacuum, they interact with the atoms and molecules of the medium. This interaction leads to phenomena such as slowing down, dispersion, and energy absorption, which can be mathematically modeled by treating the material as a homogeneous, isotropic, and linear dielectric. In this post, we provide an exhaustive derivation of the damped wave equation and explore how conductivity leads to the concepts of skin depth and complex refractive indices. [INSERT DIAGRAM OF ELECTROMAGNETIC WAVE IN A CONDUCTING MEDIUM SHOWING ATTENUATION HERE] Electromagnetic Waves in a Dielectric Medium: Theory and Derivations 1. Nature of the Medium and Constitutive Relations To analyze wave propagation in matter, we simplify the material properties. We assume the medium is Homogeneous (properties do not change with position), Isotropic (properties are the same in all directions), and Linear (the response o...

Polarisation describes the geometric orientation of the oscillations of an electromagnetic wave. While the direction of propagation tells us where the wave is going, polarisation tells us how the electric field vector behaves as it travels. In this post, we will derive the general equation of polarisation by superposing two orthogonal components and classify the resulting waves into linear, circular, and elliptical states. Polarisation of Electromagnetic Waves: Derivation from Superposition To understand the nature of polarisation, we analyze the superposition of two independent plane waves traveling in the same direction but oscillating in mutually perpendicular directions. 1. Mathematical Setup Consider a monochromatic electromagnetic wave propagating along the positive \(z\)-axis. The wave vector is \(\mathbf{k} = k\hat{z}\). Since electromagnetic waves are transverse, the electric field vector \(\mathbf{E}\) must lie in the \(xy\)-p...

How is energy transported through empty space from the sun to the earth? The answer lies in the Poynting vector, a mathematical construct that quantifies the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. In this post, we will rigorously derive Poynting's Theorem starting from the work-energy principle and Maxwell's equations. [INSERT DIAGRAM OF POYNTING VECTOR AND FIELD DIRECTIONS HERE] [Image of electromagnetic wave energy flow] Derivation of the Poynting Vector and Poynting's Theorem To derive the Poynting vector, we begin by considering the work done by electromagnetic fields on a distribution of charge. We will link the mechanical work done on charges to the decrease in energy stored within the electromagnetic fields. 1. The Rate of Work Done by Fields Consider a volume \( V \) containing a charge density \( \rho \) and current density \( \mathbf{J} \). T...