ELESSAR TELKONT
Sep23-09, 02:48 PM
1. The problem statement, all variables and given/known data
A charge distribution has the shape of a very large disk of thickness 3d. This disk has three layers of uniform densities \rho_{1},\rho_{2},\rho_{3} and thickness d each one. Within the layer with density \rho_{2} exists a tiny cubical cavity in such a manner that two of its faces are parallel to the interfaces between layers. Find the electrostatic field at the center of the cubical cavity.
2. Relevant equations
3. The attempt at a solution
I have tried to solve it basically in two manners. Directly calculating the potential due to the disk layers and using the fact that
\varphi(x)=\frac{1}{4\pi\epsilon_{0}}\int_{V}\frac {\rho}{\left\vert r-r'\right\vert}dV'+\frac{1}{4\pi}\int_{S}\left [ -\phi \frac{r-r'}{\left\vert r-r'\right\vert^{3}}+\frac{\nabla '\phi}{\left\vert r-r'\right\vert}\right ]\cdot n' dS'. My problem in both strategies is that how can I calculate the potential due to the layer where the cubical cavity is and in the second form is how to calculate the normal derivative of potential.
Is there any technic to do this problem easier?
A charge distribution has the shape of a very large disk of thickness 3d. This disk has three layers of uniform densities \rho_{1},\rho_{2},\rho_{3} and thickness d each one. Within the layer with density \rho_{2} exists a tiny cubical cavity in such a manner that two of its faces are parallel to the interfaces between layers. Find the electrostatic field at the center of the cubical cavity.
2. Relevant equations
3. The attempt at a solution
I have tried to solve it basically in two manners. Directly calculating the potential due to the disk layers and using the fact that
\varphi(x)=\frac{1}{4\pi\epsilon_{0}}\int_{V}\frac {\rho}{\left\vert r-r'\right\vert}dV'+\frac{1}{4\pi}\int_{S}\left [ -\phi \frac{r-r'}{\left\vert r-r'\right\vert^{3}}+\frac{\nabla '\phi}{\left\vert r-r'\right\vert}\right ]\cdot n' dS'. My problem in both strategies is that how can I calculate the potential due to the layer where the cubical cavity is and in the second form is how to calculate the normal derivative of potential.
Is there any technic to do this problem easier?