Surface charge density of a plane

[Guillem_dlc]

Homework Statement

A positive point charge of value $q=+2\, \mu \textrm{C}$ is located at coordinate point $(0,0,5)\, \textrm{cm}$, above an infinite homogeneously charged plane located at $z=0$. If the flux through a sphere of radius $R=10\, \textrm{cm}$ centered at the origin of coordinates is $1,08\cdot 10^5\, \textrm{Nm}^2/\textrm{C}$, calculate the value of the surface charge density of the plane.

Relevant Equations

Gauss's Law

$$\phi_E=\dfrac{Q_{\textrm{enclosed}}}{\varepsilon_0}\Rightarrow Q_{\textrm{enclosed}}=9,6\cdot 10^{-7}\, \textrm{C}$$
$$Q_{\textrm{enclosed}}=\sigma S=\sigma \pi R^2\Rightarrow \sigma =\dfrac{Q_{\textrm{enclosed}}}{\pi (0,1^2)}=3,04\cdot 10^{-5}\, \textrm{C}/\textrm{m}^2$$

I have a lot of problems with the flux exercises. I have a hard time seeing how they act when I have more than one element in the system as in this case.

 

[Gordianus]

Can you draw the point charge, the plane and the sphere?

 

[Guillem_dlc]

Can you draw the point charge, the plane and the sphere?

 

[Orodruin]

View attachment 298771

You have drawn the point charge outside of the sphere. Is this accurate according to the problem formulation?

 

[Guillem_dlc]

You have drawn the point charge outside of the sphere. Is this accurate according to the problem formulation?

No, I see now. I think that I have the solution:

$$\phi_E = \dfrac{Q_{\textrm{enclosed}}}{\varepsilon_0} \Rightarrow Q_{\textrm{enclosed}}=9,6\cdot 10^{-7}\, \textrm{C}$$
$$\phi_{\textrm{enclosed}}=\sigma S=\sigma \pi R^2+q\Rightarrow \sigma=\dfrac{Q_{\textrm{enclosed}}-q}{S}=-33\, \mu \textrm{C}/\textrm{m}^2$$

 

[Gordianus]

A good drawing fixes many problems.