Laplace equation involving a disk

[dp182]

Homework Statement

disk has a radius C with boundary conditions
V(C,[itex]\vartheta[/itex])={cos([itex]\vartheta[/itex]) ;-[itex]\pi[/itex]/2[itex]\leq[/itex][itex]\vartheta[/itex][itex]\leq[/itex][itex]\pi[/itex]/2. 0; otherwise
solve the laplace equation

Homework Equations

c_o/2+d_o/2ln(r)+[itex]\sum[/itex](cn₁rⁿ+dn₁r^-ncos(n[itex]\vartheta[/itex])+(cn₂rⁿ+dn₂r^-nsin(n[itex]\vartheta[/itex])

The Attempt at a Solution

my main problem is where do I input my boundaries if I am not given an initial V(0,[itex]\vartheta[/itex]) do I just assume T_o and solve or do I use somethings else

 

[mighty2000]

Hello,

Thank you for your question. In order to solve the Laplace equation for this disk with the given boundary conditions, you will need to use the method of separation of variables. This involves assuming a solution of the form V(r, \vartheta) = R(r)\Theta(\vartheta) and then solving for each function separately.

For the radial function R(r), you can use the boundary conditions to solve for the coefficients in the series solution. However, for the angular function \Theta(\vartheta), you will need to use the given boundary conditions at \vartheta = -\pi/2 and \vartheta = \pi/2 to solve for the coefficients in the series solution.

Once you have the series solutions for both R(r) and \Theta(\vartheta), you can combine them to get the overall solution for V(r, \vartheta). Don't forget to include the constant term in the series solution for R(r) to account for the boundary condition V(0, \vartheta) = 0.

I hope this helps. If you have any further questions, please let me know. Good luck with your solution!