3.4.12 Problems and solutions

Throughout this section the media are simple (linear, isotropic, homogeneous) dielectrics of permittivity $\epsilon ={\epsilon }_r{\epsilon }_0$, so that

and in a charge-free region ${\nabla }^{2}V=0$. At an interface $E_{1\tau }=E_{2\tau }$ and $D_{2n}-D_{1n}={\rho }_S$ ($a_n$ from medium 1 to medium 2), and at the surface of a conductor $D_n={\rho }_S$ with the normal pointing out of the conductor. Bound charges: ${\rho }_{Sb}=P\cdot a_n$ on a surface of a dielectric (outward normal $a_n$) and ${\rho }_b=-\nabla \cdot P$ inside it. Constants (Chapter 1, Eq. (1.4.2)): ${\epsilon }_0=10^{-9}/(36\pi )\approx 8.842\times 10^{-12}$ F/m, $2\pi {\epsilon }_0=1/(18\times 10^9)=5.556\times 10^{-11}$ F/m, $4\pi {\epsilon }_0=1/(9\times 10^9)=1.111\times 10^{-10}$ F/m, $1/(4\pi {\epsilon }_0)=9\times 10^9$ m/F. Dielectric strength of air: $3$ MV/m.