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This tag is for questions relating to a non-conducting material called dielectric (or dielectric material). Dielectrics are very much important to explain various phenomena in various fields of physics like solid-state physics, optics, electronics, cell biophysics, etc.
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Charge in a layered ball
The value of $\rho_b$ is incorrect. Check the formula you're using.
Use $\vec{n}\cdot\ (\vec{D_{out}}-\vec{D_{in}})=\sigma_f$
$\vec{n}\cdot\ (\vec{P_{out}}-\vec{P_{in}})=\sigma_b$
On the respectiv …