This is wrong but I can't figure out what is wrong, could someone help me find the errors.find the electric magnetic field at point (x,y,z)=(x0,0,0) if surface charge density is ps=(theta', phi') =1000c/m^2 is on the surface of a sphere of radius r0 where r0<x0.

  • \$\begingroup\$ The electric field at distance X is just an inverse function of \$X^2\$ isn't it? hyperphysics.phy-astr.gsu.edu/hbase/electric/elesph.html \$\endgroup\$ – Andy aka Feb 11 '17 at 20:03
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    \$\begingroup\$ It would help a lot if you could write the question. Rho is given as surface charge, but you treat it as volume charge. What is it? \$\endgroup\$ – sweber Feb 11 '17 at 23:40
  • \$\begingroup\$ \$\bar r^{'}\$ does not always line up with \$\bar r\$, so you cannot assign the same \$\bar a_r\$ to both of them. \$\endgroup\$ – rioraxe Feb 12 '17 at 0:22

What's missing here is the original question. But I guess it is something like

For a hollow sphere with radius \$r_0\$ and surface charge \$\rho_s=1000C/m^2\$ located at the origin of the coordinate system, find the electric field at a point \$\vec{x}_0\$ outside the sphere.

In general, you have two issues in your calculation.

First, it is a surface charge, and \$dq\$ does not depend on any \$dr\$, just \$r^2_0\$:



So you'll only have to integrate over this two angles, too:


Second each small charge \$dq\$ located at a position \$dr\$ creates a tiny field at \$\vec{x}_0\$. Direction and strength depend on the distance between both:


Of course, \$r\$ is different for each \$dq\$, and since you have chosen spherical coordinates, you have to express this in spherical coordinates, too:

$$\vec{r}=\begin{pmatrix}r\,\sin \theta \,\cos \varphi \\r\,\sin \theta \,\sin \varphi \\r\,\cos \theta \end{pmatrix}$$

With this, the integral is



Finally, should you really calculate the field this way? The integral is a bit nasty... Typically, one uses Gauß law plus geometry to get the well known result



$$q=4\pi r_0^2\rho_s$$

EDIT: Lost the \$\sin\theta\$...


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