A hollow metallic sphere has inner and outer radii $a$ and $b$ respectively. How to calculate its resistance between two a points $A$ (on the inner surface) and a point $B$ (on the outer surface)? The resistivity of the metal is $r$.
What is the simplest approach to this problem?Is it possible to calculate without using integration?How to apply integration to this problem? Please help!!
Best Answer
Since Michael has already pointed out that the problem as stated has no answer, I will answer a different question instead: if we have a resistive spherical shell with inner radius $a$, outer radius $b$, and bulk resistivity $\rho$, and the surfaces of this shell are coated with a conductive layer, what is the resistance between the inner and outer surface?
Now we can break the problem into a simple integration. Imagine the shell to be made up of infinitesimally thick shells at radius $r$ and with thickness $\delta r$. The same current $I$ has to flow through each consecutive shell (conservation of charge) so we should be able to compute the resistance of this shell. As all the shells are in series, we can integrate the expression to give the total resistance. Here we go:
Area of shell:
$$A = 4\pi r^2$$
Resistance
$$\delta R = \frac{\rho \delta r}{A} = \frac{\rho \delta r}{4\pi r^2}$$
Total resistance:
$$R = \int_a^b \frac{\rho dr}{4\pi r^2} = \frac{\rho}{4\pi}\left(\frac{1}{a}-\frac{1}{b}\right)$$
Incidentally you can see that if $a$ goes to zero, this resistance becomes infinite - just as it does when the contact surface of the point is truly a "point" as it appears to be in your question.