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Suppose in a given situation, a dielectric extends in all space till infinity. Now there is a cavity of radius $R$ centered at origin. At the origin, there is a point charge $Q$.

The question asks us to find the net bound surface charge. While finding this, should I consider the charge that would supposedly appear at infinity too (is it defined at all?), which would make the answer zero, or should it be the non-zero value that would appear at the inner surface of the cavity in the dielectric?

(The question was worded specifically to include "net", so my real question is whether any bound charge can be defined in this situation except that at the inner surface)

garyp
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pulsejet
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3 Answers3

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What's the electric field at infinity?

$$\vec{E}=\frac{1}{4\pi\epsilon}\frac{Q}{r^2}\hat{r}|_{\substack{r=\infty}}=0$$ Polarization=$$\vec{P}=\epsilon_0\chi_e\vec{E}=0$$

Bound surface charge=

$$\sigma_b=\vec{P}\cdot{\hat{n}}=0$$

So no bound surface charge at infinity.

Consider only the charge at the surface of the cavity.

UPDATE:

As suggested by @Radial Apps I'm putting this (useful) link that explains dielectrics. This is the link from internet archive.

GeeJay
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The problem is probably asking for only the induced charge on the surface of the cavity. After all, what would be the point in detailing the configuration if the answer is going to be $0$ regardless? In addition, it's also difficult to talk about bound charges at infinity. The dielectric extends in all space so there's no identifiable outer boundary at which charges can be induced.

The problem itself can be solved by integrating $\nabla\cdot\bf{D}=\rho_{\mathrm{free}}$ over a sphere that includes the entire cavity to obtain an expression for $\bf{D}$. Assuming the permittivity of the dielectric is a scalar $\epsilon$, you can then write $\bf{E}=\bf{D}/\epsilon$. By noting that $\bf{E}$ must be equal to the electric field produced by both the charge at the center and the charges induced on the surface of the cavity, you should be able to find an expression for the total induced charge.

honey.mustard
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There is nothing special about the bound charge except that it is derived only from the boundary of the dielectric in this particular case (volume charge density of bound charges, that is bound charge inside the dielectric is zero here). As long as a boundary is defined so is the bound charge. Therefore the net charge is zero.

SAKhan
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