# Maxwell's Equations

#### tiagobt

I'm trying to understand the physical meaning of Maxwell's Equation, but I'm confused about what generates what. According to Gauss's Law, electric charge placed somewhere generates electric flux, whereas Gauss's Law for Magnetism says that charge itself doesn't generate magnetic field. Faraday's Law says that magnetic field changing in time generates electric field, which may also generate voltage. Ampère's Law says that current (charge changing in time) generates magnetic field. Is this right?

Can electric field also generate magnetic field? How can I see this through Maxwell's Equations? How can I apply these concepts to the propagation of electromagnetic waves?

Thanks,

Tiago

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#### mikeu

The relevant Maxwell's equations in a vacuum are:

$$\nabla\times \mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}$$ (Faraday's law)

$$\nabla\times\mathbf{B} = \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}$$ (Ampere's law)

So from Faraday's law you can see how a changing magnetic field can affect the electric field through the time derivative of B, while Ampere's law shows that a changing electric field affects the magnetic field similarly. Thus each type of field can change (or create) the other type. As for the propagation of electromagnetic waves, you want to take the curl of these equations and use some vector identities to derive the wave equation. You should be able to derive, for example,

$$\nabla^2\mathbf{E}=\mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}$$

which is of course of the form of the wave equation, say for a field A with velocity v:

$$\nabla^2\mathbf{A}=\frac{1}{v^2}\frac{\partial^2\mathbf{A}}{\partial t^2}.$$

So the E field in a vacuum is a wave moving at speed $$c=\frac{1}{\sqrt{\mu_0\epsilon_0}}$$,

Mike

#### trytohelp

Basics first.
An electric field originates with a charged particle, only when the charged particle is in motion can it create a magnetic field. Then based on its mass and velocity it creates a matter wave equation, or if it occillates to create a wave phenomonon. These create the wavelength, the velocity of propogation is given in Mikeus' reply.

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