An electret bead used in a dust sensor is a sphere of radius with a frozen-in

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electric fields in matter · Electric fields in matter

The Boundary-Value Problem, Ch Polarisation and bound charge, completed

Problem

An electret bead used in a dust sensor is a sphere of radius a=4.0 mma = 4.0\,\mathrm{mm} with a frozen-in, uniform polarisation P=Pz^\mathbf{P} = P\hat{\mathbf{z}}, P=5.0×10−6 C m−2P = 5.0\times10^{-6}\,\mathrm{C\,m^{-2}}. It sits in vacuum with no free charge anywhere. ((a)) Let a point charge qq sit at r′\mathbf{r}' inside an imaginary sphere of radius R>∣r′∣R > |\mathbf{r}'| centred at the origin. Using the antisymmetry of the Coulomb field in its two position arguments, relate ∫r<REq(r) dτ\int_{r<R}\mathbf{E}_q(\mathbf{r})\,d\tau to the field, at the point r′\mathbf{r}', of a fictitious uniformly charged ball of total charge qq filling the same sphere; then use Gauss's law for that ball to show ∫r<REq dτ=−qr′3ε0.\int_{r<R}\mathbf{E}_q\,d\tau = -\frac{q\mathbf{r}'}{3\varepsilon_0}. ((b)) Sum this result over any set of point charges lying inside the sphere of radius RR to show that ∫r<RE dτ=−p/3ε0\int_{r<R}\mathbf{E}\,d\tau = -\mathbf{p}/3\varepsilon_0, where p\mathbf{p} is their total dipole moment about the origin, whatever the value of RR (provided it encloses all the charges). ((c)) Derive σb=P⋅n^\sigma_b = \mathbf{P}\cdot\hat{\mathbf{n}} and ρb=−∇⋅P\rho_b = -\nabla\cdot\mathbf{P} from the potential of a continuous dipole distribution, and show that for the bead ρb=0\rho_b = 0 and σb=Pcos⁡θ\sigma_b = P\cos\theta; use the resulting Legendre series and the interface conditions on VV to justify that Ein\mathbf{E}_{\rm in} inside the bead must be uniform. Then use part (b), with the bound charges as the source, to obtain Ein\mathbf{E}_{\rm in} without solving the Legendre coefficients numerically, and check the result against Ein=−P/3ε0\mathbf{E}_{\rm in} = -\mathbf{P}/3\varepsilon_0. ((d)) Give numerical values for the bead's dipole moment pp, its interior field Ein\mathbf{E}_{\rm in}, and ∫E dτ\int\mathbf{E}\,d\tau over its volume.

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