A variable capacitor used to tune a circuit has two semicircular plates of radius…

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

The Boundary-Value Problem, Ch Energy and force in dielectrics

Problem

A variable capacitor used to tune a circuit has two semicircular plates of radius

R = 5.0\,\mathrm{cm}

separated by a gap

d = 1.0\,\mathrm{mm}

; a dielectric of relative permittivity

\varepsilon_r = 2.5

fills the overlapping wedge of angle

\theta

. A battery holds the plates at a fixed potential difference

V = 800\,\mathrm{V}

while

\theta

can be varied by rotating one plate. Take the overlap angle to be

\theta = \pi/3\,\mathrm{rad}

. ((a)) The overlapping area of the wedge is

\tfrac12R^2\theta

, so

C(\theta)=\varepsilon_0\varepsilon_rR^2\theta/(2d)

. Derive, from the energy balance for a conductor held at fixed potential, the general relation between the torque tending to increase

\theta

and

(\partial W/\partial\theta)_V

, and use it to find the torque at

\theta=\pi/3

. ((b)) Verify, by evaluating

\tfrac12\int\mathbf{D}\cdot\mathbf{E}\,d\tau

directly over the dielectric-filled wedge, that this equals

W_{\rm free}=\tfrac12C(\theta)V^2

\theta=\pi/3

. ((c)) Treat one molecule of the dielectric as a charge on a spring with

\mathbf{p}=\alpha\mathbf{E}

\alpha = 2.0\times10^{-40}\,\mathrm{C\,m^2\,V^{-1}}

, sitting in the field acting on it,

E=V/d

, inside the wedge. Find the total energy of this induced dipole in the field, showing how the factor of

\tfrac12

arises.