Derive the sphere response factor a second time

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

The Boundary-Value Problem, Ch Linear dielectrics and the boundary-value problem with matter in it

Problem

Derive the sphere response factor

(\varepsilon_r-1)/(\varepsilon_r+2)

a second time, without the Legendre series: model the sphere as producing, on itself, the depolarising field

-\mathbf P/3\varepsilon_0

found for the uniformly polarised sphere, and combine this with the linear relation

\mathbf P = \varepsilon_0(\varepsilon_r-1)\mathbf E_{\rm in}

to solve self-consistently for

\mathbf E_{\rm in}

in terms of the applied field

\mathbf E_0

. ((a)) Show that

\mathbf E_{\rm in}=\dfrac{3}{\varepsilon_r+2}\mathbf E_0

, and hence write down

\mathbf p=4\pi\varepsilon_0a^3\dfrac{\varepsilon_r-1}{\varepsilon_r+2}\mathbf E_0

. ((b)) Apply the result to a mineral-oil sphere of radius

a = 5.0\,\mathrm{mm}

and

\varepsilon_r = 2.5

in a uniform field

E_0 = 8.0\times10^{4}\,\mathrm{V\,m^{-1}}

: find

E_{\rm in}

and

p

. ((c)) The point-charge image above a dielectric half-space carries the analogous denominator

\varepsilon_r+1

in place of

\varepsilon_r+2

. Writing each denominator in the form

\varepsilon_r-1+1/N

, state the value of

N

implied by the sphere and by the half-space, and say in one sentence why the half-space's

N

cannot be read as the depolarising factor of a body.