Surface charge density of a plane

• Guillem_dlc
In summary, the conversation discusses a problem involving flux and multiple elements in a system. The solution involves calculating the enclosed charge and using it to find the surface charge density. The conversation also mentions the importance of a good drawing in solving such problems.

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.

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

Gordianus said:
Can you draw the point charge, the plane and the sphere?

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

Orodruin said:
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:

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

A good drawing fixes many problems.