I have the following problem:
The temperature at a point in a cylinder of radius $a$ and height $h$, and made of material with conductivity $k$, is inversely proportional to the distance from the centre of the cylinder. Find the heat flow across the curved surface of the cylinder.
The solution says that
$T = \dfrac{\alpha}{\sqrt{x^2 + y^2 + z^2}}$ where $\alpha$ is a constant
and hence
$\mathbf{F} = \dfrac{\alpha k(x, y, z)}{(x^2 + y^2 + z^2)^{3/2}}$
$= \dfrac{\alpha k(r\cos(\theta), r\sin(\theta), z)}{(r^2 + z^2)^{3/2}}$
How and why did we go from $\sqrt{x^2 + y^2 + z^2}$ to $(x^2 + y^2 + z^2)^{3/2}$? This isn’t explained in the solution.
In $T = \dfrac{\alpha}{\sqrt{x^2 + y^2 + z^2}}$, presuming $\sqrt{x^2 + y^2 + z^2}$ is meant to be the distance from the centre of the cylinder, shouldn’t we have $z = 0$ such that we get $\sqrt{x^2 + y^2}$ as the distance from any point to the centre of the cylinder? This is because the formula for the distance between any two points in a cylinder (and indeed any object in $\mathbb{R}^3$) is $d(P_1, P_2) = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2}$. Since we want the distance to the centre of the cylinder, we would have $z_2 = z_1$ such that $z_2 – z_1 = 0$. So shouldn’t it actually be $T = \dfrac{\alpha}{\sqrt{x^2 + y^2}}$ where $\alpha$ is a constant?
I would greatly appreciate it if people could please take the time to clarify this.
In reference to your first point, by Fourier Law the heat flux $q$ is proportional to the temperature gradient. Due to simmetry, the flux is the same along any point of the outer surface, at fixed $z$. We can conveniently choose a point where the gradient simplifies to a single component, and then
$$ \lvert F \rvert = – k \frac{\partial T}{\partial x} = – k\, (-\frac{1}{2}) (x^2 + y^2 + z^2)^{-3/2} 2 x $$
Evaluating at $x=a$, and incorporating all the constants in the constant $\alpha$ (without changing its name, as its arbitrary anyhow) you get the stated result.
In referece to your second point, it seems to me perfectly valid. I wonder if they actually meant the center as the point $(0 , \frac{h}{2})$ in a cylindrical coordinate system with the origin coincinding with the center of one of the bases,and the $z$ axis aligned to the axis, directed towards the other base. The center has to be a point: otherwise, they would have maybe used the expression, “distance from the axis”.