# Induced/bound charges in conductors and dielectrics

feynman1
When placed in an electric field, a conductor has induced charges and a dielectric has bound charges. When there's no net bound charge density in the bulk of the dielectric, bound charges stay on the surface only, like induced charges in conductors. In Maxwell's eqs, the induced charges are categorized as free charges but bound charges aren't.
When a dielectric has an infinite dielectric constant, the dielectric degenerates into a conductor. But can the bound charges also be degenerated into induced charges?

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But usually the induced surface charge for conductors are not treated as free charges but free charges is extra charge you put in addition to the charges of the matter constituents. The induced surface charge is usually lumped into ##\vec{D}## and can be calcualted as the jump of the normal component of ##\vec{D}## along the surface after the boundary-value problem is solved. Of course you can always shuffle between charges and fields. All that counts at the end is the total electromagnetic field and the total charge-current distribution.

feynman1
But usually the induced surface charge for conductors are not treated as free charges but free charges is extra charge you put in addition to the charges of the matter constituents.
I don't get it. The Maxwell's eqs or Gauss' law on the boundary will show that the induced charges on conductors should appear as free charges, otherwise these eqs give wrong answers.

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What I had in mind is a problem like a conducting sphere of radius ##a## around the origin in an asymptotically homogeneous electric field, i.e.,
$$\vec{\nabla} \times \vec{E}=0, \quad \vec{\nabla} \cdot \vec{D}=0, \quad \vec{D}=\epsilon_0 \vec{E}.$$
Then you have the condition ##\vec{E}(\vec{r})=0## for ##|\vec{r}|<a##, ##\vec{E}(\vec{r}) \rightarrow \vec{E}_0## for ##|\vec{r}| \rightarrow \infty##. Another condition is of course that the total charge on the surface is 0 (for an insulated sphere). Here are no "free charges" involved. All that happens is that charge already present in the uncharged sphere is shifted.

The solution is found by introducing the potential
$$\vec{E}=-\vec{\nabla} \Phi.$$
Let ##\vec{E}_0=E_0 \vec{e}_z##. By symmetry the part of the potential describing the induced field can only be characterized by a vector. It should vanish at infinity and not lead to additional net charge. So it can only be a dipole field with ##\vec{p}=p \vec{e}_z##. So the ansatz for ##r>a## is
$$\Phi=-z \left (E_0-\frac{p}{r^3} \right).$$
For ##r \leq a## you have ##\Phi=0##. With this choice of the arbitrary additive constant the potential should vanish along the sphere, we have
$$E_0-\frac{p}/a^3=0 \; \Rightarrow \; p=E_0 a^3$$
and thus
$$\Phi=-E_0 z \left (1-\frac{a^3}{r^3} \right).$$
The surface charge distribution is most easily calculated in spherical coordinates,
$$\Phi=-E_0 \cos \vartheta \left (r-\frac{a^3}{r^2} \right),$$
$$\sigma=D_r=-\epsilon_0 \partial_r \Phi|_{r=a} = 3 \epsilon_0 E_0 \cos \vartheta.$$
From Gauss's Law it's clear that the total charge on the sphere is 0, but of course you can verify it easily directly
$$Q_{\text{ind}}=3 \epsilon_0 E_0 \int_{0}^{\vartheta} \mathrm{d} \vartheta \int_0^{2 \pi} \mathrm{d} \varphi a^2 \sin \vartheta \cos \vartheta =0.$$

• etotheipi
feynman1
What I had in mind is a problem like a conducting sphere of radius ##a## around the origin in an asymptotically homogeneous electric field, i.e.,
$$\vec{\nabla} \times \vec{E}=0, \quad \vec{\nabla} \cdot \vec{D}=0, \quad \vec{D}=\epsilon_0 \vec{E}.$$
Then you have the condition ##\vec{E}(\vec{r})=0## for ##|\vec{r}|<a##, ##\vec{E}(\vec{r}) \rightarrow \vec{E}_0## for ##|\vec{r}| \rightarrow \infty##. Another condition is of course that the total charge on the surface is 0 (for an insulated sphere). Here are no "free charges" involved. All that happens is that charge already present in the uncharged sphere is shifted.

The solution is found by introducing the potential
$$\vec{E}=-\vec{\nabla} \Phi.$$
Let ##\vec{E}_0=E_0 \vec{e}_z##. By symmetry the part of the potential describing the induced field can only be characterized by a vector. It should vanish at infinity and not lead to additional net charge. So it can only be a dipole field with ##\vec{p}=p \vec{e}_z##. So the ansatz for ##r>a## is
$$\Phi=-z \left (E_0-\frac{p}{r^3} \right).$$
For ##r \leq a## you have ##\Phi=0##. With this choice of the arbitrary additive constant the potential should vanish along the sphere, we have
$$E_0-\frac{p}/a^3=0 \; \Rightarrow \; p=E_0 a^3$$
and thus
$$\Phi=-E_0 z \left (1-\frac{a^3}{r^3} \right).$$
The surface charge distribution is most easily calculated in spherical coordinates,
$$\Phi=-E_0 \cos \vartheta \left (r-\frac{a^3}{r^2} \right),$$
$$\sigma=D_r=-\epsilon_0 \partial_r \Phi|_{r=a} = 3 \epsilon_0 E_0 \cos \vartheta.$$
$$Q_{\text{ind}}=3 \epsilon_0 E_0 \int_{0}^{\vartheta} \mathrm{d} \vartheta \int_0^{2 \pi} \mathrm{d} \varphi a^2 \sin \vartheta \cos \vartheta =0.$$