What is the intuition for no resistance between concentric spherical surfaces if $k = 0$ in this problem? [closed]

Question

I've currently solved problem 5.16 in Cheng's book on Field and Wave electromagnetics. The problem is stated as:

Determine the resistance between two concentric spherical surfaces of radii $R_1$ and $R_2$ assuming that the material has conductivity of $\sigma = \sigma_0(1+k/R)$ fills the space between them.

I managed to solve it and end up with the expression given by

$$ R = \frac{\ln\left(\frac{R_2(R_1+k)}{R_1(R_2+k)}\right)}{4\pi\sigma_0 k}.$$

After I got my solution, I wanted to do a sanity check to confirm that it's indeed reasonable.

We can see that if we increase the conductivity $\sigma_0$ then the resistance decreases, as expected.

However, if we set $k= 0$, meaning the conductivity is constant between the two conectric spherical surfaces, the resistance is $0$ since $\ln(1) = 0$. I'm very new into the concepcts of conductivity, and so I'd appreciate if anyone could give me a description of why this result would be resonable.

Answer 1

There's also a $k$ in the denominator of your expression, so simply plugging in $k = 0$ isn't valid; you would have a quantity of the form $0/0$, which is indeterminate.

Rather, you need to take the limit of $R$ as $k \to 0$ to make sense of this. If you do, you will confirm that the resistance approaches a non-zero value (assuming that $R_1 \neq R_2$). And if the result for the resistance between two concentric spherical shells is in your textbook, your limit here should agree with that.