Induced Magnetic Field in Dielectric

[Hypnotoad]

I need a little help on an E&M problem I'm working on. A dielectric slab suspended in free space has a time dependent, non-uniform electric field inside of it (it was given in the problem, but I don't have it with me right now). For the material $$\mu=\mu_0$$ and $$\epsilon=2.56\epsilon_0$$. I need to first find the magnetic field inside the material and then the magnetic and electric fields just above and below the material.



For the first part, I tried using curl of E$$=-\frac{\partial{B}}{\partial{t}}$$ which gives me the partial time derivative of the magnetic field. Can I just integrate this to get the magnetic field? Since it is a partial derivative of B, I wasn't sure if that would be allowed.



For the second part, how does the electric field change ouside of the material? Does the permitivity affect the magnetic or the electric field, or both? And could someone explain the difference between permitivity and permeability?

E&M is a very weak subject for me, so expect to get a lot of questions this semester.

 

[clive]

The equation you try to apply is useless in this problem because it represents the Faraday law (for electromagnetic induction: variation of B => circular electric field). You have an inverse problem: variation of E => circular magnetic field, so...

I propose another Maxwell's equation:

curl B=miu_0*j+miu_0*eps*dE/dt

You do not have any currents, so j=0 and

curl B=miu_0*eps*dE/dt

If you replace B an E with sinusoidal functions (with the form A_0*exp[i*(k*r-w*t))] you'll find a relation between the magnitudes of the two alternative fields (electric and magnetic):

B_0=v*E_0

where v=1/sqrt(miu_0*eps)

or simpler (with your data)

v=c/sqrt(2.56)

(c-velocity of light in vacuum)

The magnetic and electric fields are allways perpendicular.

In vacuum you have the same problem but eps=eps_0.

 

[wais]

Hi there,

I'm happy to help with your E&M problem. Let's start by addressing your first question about finding the magnetic field inside the dielectric slab. You are correct in using the curl of the electric field to find the magnetic field, but you cannot simply integrate it to get the magnetic field. This equation is known as Faraday's law and it tells us that the time derivative of the magnetic field is equal to the curl of the electric field. In order to solve for the magnetic field, you will need to use the appropriate boundary conditions and Maxwell's equations.

Moving on to the second part of your question, the permittivity (ε) and permeability (μ) both play a role in determining the behavior of electromagnetic fields in a medium. The permittivity is a measure of how easily a material can be polarized by an electric field, while the permeability is a measure of how easily a material can be magnetized by a magnetic field. In general, both ε and μ affect both the electric and magnetic fields, but in different ways. For example, in your problem, the permittivity of the dielectric slab is larger than that of free space, so it will have a larger effect on the electric field than the magnetic field.

I hope this helps clarify things for you. Don't hesitate to ask more questions if you need further clarification. E&M can be a challenging subject, but with practice and a solid understanding of the fundamentals, you'll be able to tackle any problem. Best of luck with your studies!