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

The total volume of bound charge is zero.

Explanation:

We have to the volume and surface bounded charge densities.

   ρb = - Δ . p = - Δ .k ( + +)

                      = - 3k

 On the top of the cube the surface charge density is

                     σb = p . z

                           =

By symmetry this holds for all the other sides. The total bounded charge should be zero

        Qtot = (-3k)a³ + 6 . . a² = 0

               σb = -3K σb =

            Qtot = 0

Hence,  the total volume of bound charge is zero.

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explanation: