Electric field inside a polarized dielectric sphere

[Saptarshi Sarkar]

Homework Statement

A dielectric sphere of radius R carries a polarization $P = kr^2\hat r$ where k is a constant and r is the distance from the origin. Then the electric field inside the sphere is?

Relevant Equations

$ρ_b = -\nabla .P$

My attempt:

I know from Gauss' law in dielectric

$\nabla .D = ρ_f$
where $D = ε_0E + P$,

so as
$ρ_f = 0$ (as there is no free charge in the sphere)
=> $\nabla .D = 0$
=> $ε_0\nabla .E = \nabla .P$

from this I get

$E = \frac {-kr^2 \hat r} {ε_0}$

But, I know that for a uniformly polarization dielectric sphere, $E = \frac {-P} {3ε_0}$

Why are both the solutions not the same?

 

[Charles Link]

This one is not uniformly polarized. The polarization is a function of $r$, and it points radially outward, instead of in a single direction. $\\$ In any case, don't jump to conclusions that since $\nabla \cdot E=C \nabla \cdot P$, that $E=C P$. This often is not the case with these problems. In this case you might be right, but a homogeneous solution to $\nabla \cdot E=0$ sometimes needs to get added to the solution $E=CP$ for cases like this. Suggest you use Gauss' law and find the enclosed $\rho_p$ to check your answer.

 

[Charles Link]

This comes up a lot in both electrostatic and magnetostatic problems: $\\$
Foe electrostatics, we have $D=\epsilon_o E+P$. We have $\nabla \cdot D=\rho_{free}$. In cases where $\rho_{free}=0$, we have $\nabla \cdot D=0$. Even though we then have $\epsilon_o \nabla \cdot E=-\nabla \cdot P$, most often it is not the case that $\epsilon_o E=-P$. $\\$ A similar thing can be said for magnetostatics, where $B=\mu_o H+M$, and $\nabla \cdot B=0$. Most often, it is not the case that $\mu_o H=-M$, even though $\mu_o \nabla \cdot H=-\nabla \cdot M$. $\\$ We have a particular solution to the differential equation, but we often need to add a solution to the homogeneous equation, $\nabla \cdot E=0$, or $\nabla \cdot H=0$, to get the complete solution for $E$ or $H$.