# Infinite conductor plane near a point charge

If i have an infinite conductor plane near a point charge ( that is the configuration of the common "Method of images" example), if i calculate the conductor's inducted charged density as

$$\sigma =-\varepsilon _{0}\frac{\partial V}{\partial n}$$

that means that the electric field is only due to the conductor's inducted charged and not also due the point charge. Where am i wrong?

No, it doesn't mean the electric field is only due to the conductor's induced charge. This is so because $V$ in your expression is the resulting potential due to the point charge and the induced ones on the plane. This resulting potential gives you information about all charges that produced it. Taking its laplacian on the location of the point charge gives you $\nabla^2V=-4\pi q\delta^3(\vec{r}) /\varepsilon_0$, so it tells you a point charge also produced $V$; and using the condition for the descontinuity of the potential's normal derivative on the condutor's surface gives you the induced surface charge $\sigma$, as in your expression, which is indeed one of the sources of such $V$.
• Remember where the formula of your post comes from. It actually tells you the difference of the potential's normal derivative equals to the surface charge. Therefore, the surface charge is responsible to a discontinuity on $E_\bot = -\partial V/\partial n$ instead of being responsible to $E_\bot$ itself. – andrehgomes Nov 23 '17 at 11:29
• Imagine there are two regions in your problem. One outside the condutor and another one inside it. The formula in your post comes from the condition $E_\bot^\text{outside}-E_\bot^\text{inside}=\sigma/\varepsilon_o$ evaluated at the interface between the two regions, i.e., at the conductor's surface. Because it is electrostatics, the electric field inside the conductor vanishes, remaining only $E_\bot^\text{outside}= -\partial V/\partial n=\sigma/\varepsilon_o$. Therefore, the function $E_\bot$ is discontinuous across the two regions because of the surface charge density. – andrehgomes Nov 23 '17 at 14:28