- #1
Saptarshi Sarkar
- 99
- 13
- 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?
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?