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Question 4.33: A dielectric cube of side a, centered at the origin, carries...

A dielectric cube of side a, centered at the origin, carries a “frozenin” polarization P = kr, where k is a constant. Find all the bound charges, and check that they add up to zero.

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P =k r =k(x \hat{ x }+y \hat{ y }+z \hat{ z }) \Longrightarrow \rho_{b}=-\nabla \cdot P =-k(1+1+1)=-3 k .

Total volume bound charge:  Q_{ vol }=-3 k a^{3} .

\sigma_{b}= P \cdot \hat{ n } \text {. At top surface, } \hat{ n }=\hat{ z }, z=a / 2 ; \text { so } \sigma_{b}=k a / 2 \text {. Clearly, } \sigma_{b}=k a / 2 on all six surfaces.

Total surface bound charge: Q_{ surf }=6(k a / 2) a^{2}=3 k a^{3} . Total bound charge is zero.

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