How to apply Gauss's law when dielectric constant in a capacitor varies continuously?

Question

Let's say the problem is to compute the capacitance of a cylindrical capacitor with a dielectric material between the plates. The capacitor is considered to be long enough to ignore end effects. The inner and outer radii are $r_i$ and $r_o$ respectively and the dielectric constant varies linearly : $\epsilon = kr$, with $r$ going from $r_i$ to $r_o$ and $k$ is some constant that maintains dimensional consistency.

Assuming cylindrical symmetry of the electric field, we could have a volume for application of Gauss's law that is also cylindrical and has the same axis as the axis of the capacitor. Its radius is $r$ that is somewhere between $r_o$ and $r_i$. Let's say that the charge on the capacitor plates is q each.

When applying Gauss's law to this set up, should the electric flux over the cylindrical surface be equal to $q/kr_i$ or should it be equal to $q / kr$, where $r$ is the radius of the cylindrical volume being used for computation of the field.

As a broader question, when permittivity is mentioned in Gauss's law, does it mean permittivity at the boundary of the volume over which the electric field flux is being computed ?

Answer 1

Gauss law: $$\nabla\cdot\mathbf D = \rho \quad\quad\Longleftrightarrow\quad\quad \int\mathbf D(\mathbf r)\cdot d\mathbf S = Q$$

Transforming to E-Fields: $$\nabla\cdot(\epsilon\mathbf E) = \rho \quad\quad\Longleftrightarrow\quad\quad \int\mathbf \epsilon(\mathbf r)\mathbf E(\mathbf r)\cdot d\mathbf S = Q$$

To answer your question, now the surface integral needs to consider the explicit dependence on the dielectric constant in space. That's the integral you would need to solve. So, just choose a suitable surface, as you would, and compute the D-Flux, and then find all D-Fields, and in the end, convert everything to a E-Fields.

Or.. you can also use the differential law, assuming that it has no angular dependence, or z-dependence, just use divergence in cylindrical coordinates to solve for $E_r$ directly: $$ \nabla\cdot\mathbf D = \frac{1}{r}\frac{\partial}{\partial r}\left(rD_r\right) = \frac{1}{r}\frac{\partial}{\partial r}\left(r\epsilon E_r\right) = \rho$$

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