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Maxwell’s Equations

Tuesday, September 22nd, 2009

How to Solve Maxwell’s Equations

Description

A detailed tutorial on the solving of Maxwell’s Equations. Step by step tutorial including several examples of how to solve Maxwell’s Equations for reference.

Overview

Maxwell’s equations are a set of four partial differential equations that describe the properties of electric and magnetic fields and relate them to their sources, charge density and current density. The result of these equations is that they show light is an electromagnetic wave. The four different equations and the way to express them is as follows:

Gauss’s Law: relates electric charge contained within a closed surface to the surrounding electrical field.

Differentiation: $\nabla \cdot \mathbf{D} = \rho_f$

Integration: $\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf D\;\cdot\mathrm{d}\mathbf A = Q_{f}(V)$

Gauss’s Law for Magnetism: states that the total magnetic flux through a closed surface is zero.

Differentiation: $\nabla \cdot \mathbf{B} = 0$

Integration: $\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf B\;\cdot\mathrm{d}\mathbf A = 0$

Maxwell-Faraday Equation: describes how a changing magnetic field can create an electric field.

Differentiation: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$

Integration: $\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l}� = - \frac {\partial \Phi_{B,S}}{\partial t}$

Ampere’s Circuital Law: states that magnetic fields can be generated by electrical current and changing electric fields.

Differentiation: $\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t}$

Integration: $\oint_{\partial S} \mathbf{H} \cdot \mathrm{d}\mathbf{l} = I_{f,S} + \frac {\partial \Phi_{D,S}}{\partial t}$

Tags: Ampere's Circuital Law, change density, closed surface, current density, density, electric charge, electric fields, electrical current, electromagnetic wave, Gauss's Law, Gauss's Law for Magnetism, Guassian surface, light, magnetic field, magnetic flux, Maxwell's equations, Maxwell-Faraday Equation, Physics, Science, zero
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Telegrapher’s Equations

Friday, September 18th, 2009

How to Solve Telegrapher’s Equations

Description

A detailed tutorial on the solving of Telegrapher’s Equations. Step by step tutorial including several examples of how to solve Telegrapher’s Equations for reference.

Overview

Telegrapher’s Equations, sometimes referred to simply as telegraph equations, are a pair of differential equations which meausre the voltage and current on a transmission line with regard to distance and time. An example would be a telegraph, hence the name. Instead of having an actual set of equations, Telegrapher’s Equations tend to more oftenbe expressed as a schematic, with the equations only being used for things such as loops and transmissions.

Tags: attenuation constant, differential equations, distance, loops, magnetic field, pair, phase constant, Physics, primary line constants, propagation constant, Science, telegraph, telegraph equations, telegrapher's equations, time, transmission
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Posts Tagged ‘magnetic field’

Maxwell’s Equations

How to Solve Maxwell’s Equations

Telegrapher’s Equations

How to Solve Telegrapher’s Equations