# Electric field-boundary conditions question

1. Jan 16, 2017

### Abdulwahab Hajar

1. The problem statement, all variables and given/known data
Two lossy homogeneous dielectric media with dielectric constans ϵrl = 2, ϵr2 = 3 and conductivities a1= 15 (mS), σ2 = 10 (ms) are in contact at the z = 0 plane. in the Z>0 region a uniform electric field E1 = 20i - 50k
exists (i and k being unit vectors in the x and z directions)..
Ok so my question is this in one of the parts of the question it asks for the surface charge density on the interface, however since both media are dielectric shouldn't the surface charge density be 0 according to the boundary conditions of electric fields
2. Relevant equations
D1n - D2n = Ps (surface charge density)
where D1n - D2n are the normal components of the electric flux densities of media 1 and 2 respectively
3. The attempt at a solution
the way I see it, and according to what is stated in the textbook Ps should be 0, however the solution manual states differently

2. Jan 18, 2017

$\nabla \cdot D=\rho_{free}$ gives (by Gauss' law) $\int D \cdot \, dA =Q_{free} =0$ because there is no free electrical charge. The $D$ integral gives $-\epsilon_1 E_1+\epsilon_2 E_2=0$ This means that the electric field is discontinuous across the boundary. They are apparently providing you with the value of $E_1$. From the equation $\int E \cdot \, dA=\frac{Q_{total}}{\epsilon_o}=\frac{Q_p}{\epsilon_o}$ (since $Q_{total}=Q_{free}+Q_p$ with $Q_{free}=0$), you should be able to compute the surface polarization charge density $\sigma_p$. They don't supply any units on the electric field $E_1$ but I would presume you can assume units of Newtons/Coulomb=Volts/meter.