Electric Potential of a Uniformly Charged Disk of Charge — Off Axis

A disk of radius R normal to the z axis centered at the origin (i.e., lying in the x-y plane) holds a uniform charge density σ;  Find and plot Vfar and Vnear — the off-axis solutions for z > 0.

HINTS:  
(i) Treat as a 2D problem.  Any plane through the z-axis will do — take the x-z plane (i.e., we consider y = 0).  This is equivalent to saying that there is no ϕ dependence.  Since any choice of ϕ will yield the same result, we can choose ϕ = 0 (and/or π).  These choices correspond to the x-z plane.

(ii)  Use the solution for the Potential along the symmetry (z) axis as a boundary condition for the solution of Laplace’s Equation in Spherical Coordinates.

(a) Set up the series solution for z > 0 ,  r > R  (i.e., consider only the (1/(r^(l +1)))(Subscript[P, l])(Cos[θ] terms in the expansion).

<Enter your answer in this text cell>

(b) Use the facts that at θ = 0, that all of the Legendre Polynomials = 1 when θ = 0, and the hint above to find the (Subscript[B, l])’s for the desired solution.  Write Vfar[r,θ] as the sum of the first four non-zero terms of the resulting series.

<Reality check>
If we set θ = 0 what are the(Subscript[P, l])’s?   To find out, execute:   LegendreP[n, Cos[0]].

(* Input code below *)
ClearAll["`*"]  (* Leave the ClearAll statement *)
LegendreP[n, Cos[0]] 
(* For all n *)

Since n is arbitrary, the answer “1” implies that every Legendre polynomial equals one when θ = 0.  This is a property of Legendre polynomials.  Below is an example of how this works out for n = 5:  (63 - 70 + 15)/8 = 1.

LegendreP[5, Cos[θ]]  (*   just an example  *)

For θ = 0, all of the Legendre Polynomials = 1.

<Enter your derivation of the (Subscript[B, l]) values below>
<Create new code and text cells where it helps>

(c) Convert your Vfar[r,θ] to Cartesian coordinates for plotting.  Plot the result, V[x,z], using Plot3D.  You will have to choose “reasonable” values for R, σ, and ϵo.  I chose R = 1, σ = 1, and ϵo = 1.  (Computers and humans seem to do better with input values closer to 1 than (10^(-12)).)

(*Input code below *)

(d)  Now repeat the above for r < R to find Vnear[r,θ].  Keep the first six nonzero terms.

<Enter your derivation of the (Subscript[A, l]) values below>
<Create new code and text cells where it helps>

(e) Convert your Vnear[r,θ] to Cartesian coordinates for plotting.   Plot the result, V[x,z].  You will have to give R, σ, and ϵo reasonable values again.  The old ones have been erased.

(* Input code below *)

(f)  Now combine the two plots (both r < R and r > R).   Reminder: here we are considering z >0 only.

For my sanity, I copy and paste the two equations for Vfarxz[x,z] and Vnearxz[x,z] from above, not forgetting to redefine R, σ, and ϵo in terms of actual numbers (all 1).

(* Input code below *)

(g)  Interpret this Plot.

<Enter your interpretation in this text cell>

(h) Create a ContourPlot plot of the potential for points in the x-z plane.  Calculate and plot the electric field for the same region using VectorPlot.  Finally, produce a Streamline plot of the electric field (again, for the same region in the x-z plane).

(* Input code below *)

(i) Show the ContourPlot and StreamPlot together.

(* Input code below *)

(i) Interpret your ContourPlot + StreamPlot.  Where is the disk?

<Enter your interpretation in this text cell>