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

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 ?

• What formula do you use for Gauss' law?
– nasu
Jan 21, 2021 at 23:48
• Total flux through the boundary of the volume equals the enclosed charge divided by the permittivity. Jan 22, 2021 at 17:56
• So where is this q/kr coming from?
– nasu
Jan 22, 2021 at 20:53

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: $$\nabla\cdot(\epsilon\mathbf E) = \rho \quad\quad\Longleftrightarrow\quad\quad \int\mathbf \epsilon(\mathbf r)\mathbf E(\mathbf r)\cdot d\mathbf S = Q$$
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: $$\nabla\cdot\mathbf D = \frac{1}{r}\frac{\partial}{\partial r}\left(rD_r\right) = \frac{1}{r}\frac{\partial}{\partial r}\left(r\epsilon E_r\right) = \rho$$