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**1. The problem statement, all variables and given/known data**

A semi infinite slab of material, [tex]-\infty<x<0, -\infty<z<\infty, -d/2<y<d/2[/tex]

has uniform polarization P in the +y direction.

What are the E and D field along x axis at y=0.

**2. Relevant equations**

[tex]\oint \vec{D}\cdot d\vec{a} = Q_{free enclosed}[/tex]

[tex]D= \epsilon_{0}E+P[/tex]

**3. The attempt at a solution**

If I use the integral statement, there is no free charge, D is uniformly 0.

This seems uninteresting and incorrect...as well as inconsistent with the second relation.

I already know that at the x=y=0 on the boundary, the magnitude of E field drops to 1/2 its "ideal infinite capacitor" (from surface charge due to polarization). IF D were always 0 and P is uniform, than E would have to be uniform, which it is not.

So why is the integral invalid? I recall qualitatively, that the E and D field are inversely related within the material i.e. if one increased the other decreased...outside trivially they are equal, with a dielectric multiplication. How do I correctly calculate D?

Thanks in advance

**1. The problem statement, all variables and given/known data**

**2. Relevant equations**

**3. The attempt at a solution**

**1. The problem statement, all variables and given/known data**

**2. Relevant equations**

**3. The attempt at a solution**

**1. The problem statement, all variables and given/known data**

**2. Relevant equations**

**3. The attempt at a solution**

**1. The problem statement, all variables and given/known data**

**2. Relevant equations**

**3. The attempt at a solution**

**1. The problem statement, all variables and given/known data**

**2. Relevant equations**

**3. The attempt at a solution**