The following exposition and question are based on two problems from the book "Electricity and Magnetism" by Purcell and Morin (problems 4.32 and 4.6 from Chapter 4 called "Electric Currents").
Imagine that we have two graphite rods of equal length.
One is a cylinder of radius $a$ and the other is conical having a radius that tapers linearly from $a$ to $b$. Here is a depiction of this setup
Using an argument based on slicing up the conical rod into disk-like slices, it can be shown that the end-to-end resistance of the conical rod is $a/b$ times that of the cylindrical rod.
On this result, Purcell and Morin say that it
is only an approximate one, valid in the limit where the taper is slow (that is, where $a-b$ is much smaller than the length of the cone). It can't be universally valid, because in the $b\to \infty$ limit the resistance of the cone would be zero. But the object shown in the figure below certainly doesn't have a resistance that approaches zero as $b\to\infty$. Why isn't this result valid?
To cut to the chase regarding my question, it is about Purcell and Morin's answer to the question posed in the snippet above. Here is their answer
The error in the reasoning is that the current doesn't fan out uniformly in the cone. If we are to build up the cone from cross-sectional slices and then add up the resistances from these slices, all parts of a given slice must be at the same potential. Equivalently, when you write down the resistance of each slice you will be assuming that the current flows perpendicular to each slice. This is clearly not the case for the slice shown below
Questions
Why must all the slices be equipotentials?
Why is this equivalent to assuming current flows perpendicular to each slice?
I think the answer is based on the fact that in the approximation that leads to the result that $R_{conical}=\frac{a}{b}R_{cylinder}$, we use the equation
$$R=\frac{L}{\sigma A}\tag{1}$$
where $\sigma$ is the conductivity of the graphite and $L$ and $A$ are length and cross-sectional area of one of the disk slices used to approximate the conical rod.
This equation is based on a number of assumptions.
One of them is that the current passing through the conductor is steady, ie the current density is constant everywhere, and thus so is the electric field.
Now, for a cylinder, assuming current enters and leaves the rod from an external conductor that forms an equipotential at the ends of the rod, this assumption is met. Here is a depiction of what I mean
Even if we don't have a perfect situation like in (a) above, if the length of the rod is much larger than the cross-sectional area of the rod, then the "end effects" shown above in (b) are less important and the assumption is less imprecise.
So how do we reason about the conical rod?
If the taper is very slow, then we approximate the entire conical rod as a cylindrical rod and perform the calculations. In treating the conical rod like a cylinder, implicitly we are assuming that the current density and electric field are perpendicular to each slice (as they would be in a cylinder), and we assume this because each successive slice is pretty much the same as the previous one.
In addition, since in our approximation each slice is infinitesimally small, it represents an equipotential surface.
However, if we have a fast taper, then each slice is not as similar to the previous slice as in the slow taper case. The idea that the current travels perpendicular to each slice is not true anymore because of "end effects" between slices. I am not sure if this is all true, it is my current understanding.
