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## Main Question or Discussion Point

The electric field due to a dipole distribution in volume ##V'## can be viewed as electric field due to a volume charge distribution in ##V'## plus electric field due to a surface charge distribution in boundary of ##V'##.

##\displaystyle\mathbf{E}=\int_{V'} \dfrac{\rho (\mathbf{r}-\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|^3} dV'

+ \oint_{S'} \dfrac{\sigma (\mathbf{r}-\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|^3} dS'=\mathbf{E}^{V}+\mathbf{E}^{S}##

##\mathbf{E}^{S}## is discontinuous at the surface ##S'##.

(1) Does this mean that ##\mathbf{E}^{S}## is undefined at the surface ##S'##? (Consequently ##\mathbf{E}## will be undefined at the surface ##S'##)

(2) If no, then what is ##\mathbf{E}^{S}## at the surface ##S'##?

##\displaystyle\mathbf{E}=\int_{V'} \dfrac{\rho (\mathbf{r}-\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|^3} dV'

+ \oint_{S'} \dfrac{\sigma (\mathbf{r}-\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|^3} dS'=\mathbf{E}^{V}+\mathbf{E}^{S}##

##\mathbf{E}^{S}## is discontinuous at the surface ##S'##.

(1) Does this mean that ##\mathbf{E}^{S}## is undefined at the surface ##S'##? (Consequently ##\mathbf{E}## will be undefined at the surface ##S'##)

(2) If no, then what is ##\mathbf{E}^{S}## at the surface ##S'##?