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UPDATE: the first 2 assignments are done (i think). I'm stuck on 3. and I have explained in attempted solution what I have tried thus far.
I have an important homework assignment due in Electromagnetism, and I have no idea where to start. It has many subassignments, but I cannot even figure out where to begin with the first one.
1. Homework Statement
The electrostatic potential with in a nonconducting sphere, with radius R (##r = \sqrt{x^2+y^2+z^2} < R##) is given as:
where a and b are constants, and with the origin set in the center of the sphere. The sphere has a surface charge of ##\sigma##, there are no charges inside or outside the sphere, and there are no external fields. The potential has rotational symmetry around the z axis, both inside and outside, and goes to zero when ##r \rightarrow \infty##.
No idea
I haven't really a attempted a solution, as i have no idea where to start. I have considered that there might be a boundary or something that I could use, but from my reading of the book, I cannot make out what it would be (sometimes it says there are no boundary on potentials, and others times there are. Cannot differentiate between the cases). I also contemplated whether it wants me to solve the laplacian equation ##\nabla^2 \cdot V=0##, but since I already have the equation, that doesn't seem to be the way to go.
EDIT: After thinking on it a bit more; If there is rotational symmetry on around the z axis, doesn't that mean that ##a = b##, considering they are just constants that either increase or decrease the potential in either x or y direction? And in that case, wouldn't it be right to then take the laplacian equation ##\nabla^2 \cdot V=0##, as there isn't any charges inside the sphere?
UPDATE:
Using the general solution laplace equation in spherical coordinates, for the 3. assignment ##V_u(x,y,z)=\sum_{n=0}^\infty A_m*r^m+B_m*r^{m1}*P_3(\cos(\theta))## I argue that ##A_m=0## as the boundary ##V \rightarrow 0## when ##r \rightarrow \infty## wouldn't be met otherwise. But beyond that I'm stuck. I don't see how I can argue the sign change in the ##r## power.
I have an important homework assignment due in Electromagnetism, and I have no idea where to start. It has many subassignments, but I cannot even figure out where to begin with the first one.
1. Homework Statement
The electrostatic potential with in a nonconducting sphere, with radius R (##r = \sqrt{x^2+y^2+z^2} < R##) is given as:
where a and b are constants, and with the origin set in the center of the sphere. The sphere has a surface charge of ##\sigma##, there are no charges inside or outside the sphere, and there are no external fields. The potential has rotational symmetry around the z axis, both inside and outside, and goes to zero when ##r \rightarrow \infty##.
 (done) Find ##a## and ##b##
 (done) Express ##V_i## in ##r## and the polar angle ##\theta## around the ##z##axis, and show, that the ##\theta## dependence is given by a Legendrepolynomium ##P_n(cos(\theta))##
 Argue briefly, that the potentiale outside the sphere, can be written as ##V_u(x,y,z)=\sum_{n=0}^\infty B_m*r^{m1}*P_3(\cos(\theta))##
 Determine ##B_m## by exploiting the continuity of the potential across the sphere
 Show that the surface charge on the sphere is ##\sigma(\theta)=K*(5*\cos(\theta)^33*\cos(\theta)) and determine ##K##
Homework Equations
No idea
The Attempt at a Solution
I haven't really a attempted a solution, as i have no idea where to start. I have considered that there might be a boundary or something that I could use, but from my reading of the book, I cannot make out what it would be (sometimes it says there are no boundary on potentials, and others times there are. Cannot differentiate between the cases). I also contemplated whether it wants me to solve the laplacian equation ##\nabla^2 \cdot V=0##, but since I already have the equation, that doesn't seem to be the way to go.
EDIT: After thinking on it a bit more; If there is rotational symmetry on around the z axis, doesn't that mean that ##a = b##, considering they are just constants that either increase or decrease the potential in either x or y direction? And in that case, wouldn't it be right to then take the laplacian equation ##\nabla^2 \cdot V=0##, as there isn't any charges inside the sphere?
UPDATE:
Using the general solution laplace equation in spherical coordinates, for the 3. assignment ##V_u(x,y,z)=\sum_{n=0}^\infty A_m*r^m+B_m*r^{m1}*P_3(\cos(\theta))## I argue that ##A_m=0## as the boundary ##V \rightarrow 0## when ##r \rightarrow \infty## wouldn't be met otherwise. But beyond that I'm stuck. I don't see how I can argue the sign change in the ##r## power.
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