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Just a picture of my configuration.

The assumption here is $$\epsilon_a,\epsilon_b,\epsilon_c$$ are different from one another. Really the interest of this problem is to find the scalar potential $$\phi$$, such that $$\nabla^2 \phi = 0$$.

So now my question, about jump conditions,

Surface at $$y=0$$ has tangent $$\vec{E}$$ continous, thus

\begin{align}

-\hat{x} \cdot \nabla \phi_a = -\hat{x} \cdot \nabla \phi_b \\

-\hat{x} \cdot \nabla \phi_a = -\hat{x} \cdot \nabla \phi_c \\

\end{align}

However if we look at $$x=0$$ then normal $$\vec{D}$$ is continous thus

\begin{align}

-\epsilon_b \hat{x} \cdot \nabla \phi_b = - \epsilon_c \hat{x} \cdot \nabla \phi_c \\

\end{align}

From our relation above this implies that $$\epsilon_b=\epsilon_c$$, which we made no such assumption. So this looks like a contradiction to me.

Can someone tell me where I have gone wrong?

Thank you!

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# Boundary condition for electrostatics problem - found issue?

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