How to Find the Steady State Temperature Distribution in a Spherical Shell?

[Poirot1]

A metallic spherical shell occupying the region given in terms of spherical polar coordinates \( (r, \theta, \phi)\) by \(r \le a \) has its surface \(r = a\) maintained at temperature
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\(u(a,\theta)=1 + \cos(\theta)+2 \cos^2(\theta) \)


Using the general solution for the Laplace equation \( u(r,\theta)=\sum A_n r^n P_n \) where the Pn are legendre polynomials, find the (axisymmetric) steady state temperature distribution \(u(r,\theta) \) within the shell.<<there was a long incomprehensible expression here which seems to have disappeared ?>>
You may assume the legrendre polynomials (first three) and that Legendre polynomials satisfy the orthogonality relation.
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[CaptainBlack]

I'm afraid the some of your notation is incomprehensible. It is generally not possible to copy formatted mathematics into the question input box and have it render correctly.

I have changed some of your expressions to the LaTeX we use here, though some of the expressions look wrong, but there is still a large expression that makes no sense <<which seems to have disappeared not only from the post but from the edit history>>.

CB

 

[Poirot1]

Thanks CB. I did it in proper code and then my internet stopped working so I tried to copy and paste. I don't know what expression you are referring to but you have all you need.

 

[Opalg]

A metallic spherical shell occupying the region given in terms of spherical polar coordinates \( (r, \theta, \phi)\) by \(r \le a \) has its surface \(r = a\) maintained at temperature


\(u(a,\theta)=1 + \cos(\theta)+2 \cos^2(\theta) \)


Using the general solution for the Laplace equation \( u(r,\theta)=\sum A_n r^n P_n \) where the Pn are legendre polynomials, find the (axisymmetric) steady state temperature distribution \(u(r,\theta) \) within the shell.<<there was a long incomprehensible expression here which seems to have disappeared ?>>
You may assume the legrendre polynomials (first three) and that Legendre polynomials satisfy the orthogonality relation.
​

The general solution for the Laplace equation should read $$u(r,\theta) = \sum A_nr^n P_n\color{red}{(\cos\theta)} = A_0P_0(\cos\theta) + A_1rP_1(\cos\theta) + A_2r^2P_2(\cos\theta) +\ldots.$$ If you substitute the formulas for the first three Legendre polynomials (which you are allowed to assume, so presumably you ought to know them), ignore the remaining terms, and put $r=a$, then you get the equation for the surface temperature in the form $A_0P_0(\cos\theta) + A_1aP_1(\cos\theta) + A_2a^2P_2(\cos\theta) = 1 + \cos\theta + 2\cos^2\theta.$ Now all you have to do is to compare coefficients of powers of $\cos\theta$ in order to determine the values of $A_0$, $A_1$ and $A_2.$

 

[Poirot1]

Ok I did that, substitued x=acosϕsin θ, and it didn't work out. Was I meant to do that?