Dielectric boundary value problems

[lys04]

Homework Statement

Why in this problem when solving for the electric potential in the z<0 region we introduce another image charge that is the same as the original charge instead of just using the original charge? I don't really get the explanation they provide.

Relevant Equations

$$\vec{D}=\epsilon \vec{E}$$

 

[Gavran]

Hard to read.
The text should be fixed. It is not clear.

 

[TSny]

Homework Statement: Why in this problem when solving for the electric potential in the z<0 region we introduce another image charge that is the same as the original charge instead of just using the original charge?

When finding the potential $\phi_1$ in region z<0, the image charge is not the same as the original charge. The image charge has the same location as the original charge, but the charge of the image is $q''$, which differs from the original charge $q$. The value of $q''$ will be determined in the analysis.

Also, when writing the potential $\phi_1$, we imagine that the purple dielectric $\varepsilon_1$ extends throughout all of space: both z < 0 and z > 0. Thus, although $q''$ is located in the region $z > 0$, $q''$ is taken to be immersed in dielectric material with $\varepsilon_1$.
That's why you see this figure

Likewise, in writing the potential for $\phi_2$, the yellow dielectric $\varepsilon_2$ is assumed to extend throughout all of space.