JesseC
Jan21-11, 04:35 PM
Standard electrostatics problem (in spherical polar coords): spherical cavity of radius R in an infinite dielectric of permittivity ε centred at origin of the coord system. Surface charge stuck on to the cavity:
\sigma(\theta) = \sigma_0 \cos (\theta)
Problem is to find the potential in all space. Looking at a solution I don't understand where some of the boundary conditions have come from, specifically:
\left[ \frac{\partial V_2}{\partial r} - \frac{\partial V_1}{\partial r} \right] _{r=R} = - \frac{1}{\epsilon \epsilon_0} \sigma_0 \cos (\theta)
(2) and (1) refer to region outside and inside the cavity respectively.
Is this just another way of writing:
\left[ \vec{D}_2 - \vec{D}_1 \right] _{r=R} = \sigma
?
\sigma(\theta) = \sigma_0 \cos (\theta)
Problem is to find the potential in all space. Looking at a solution I don't understand where some of the boundary conditions have come from, specifically:
\left[ \frac{\partial V_2}{\partial r} - \frac{\partial V_1}{\partial r} \right] _{r=R} = - \frac{1}{\epsilon \epsilon_0} \sigma_0 \cos (\theta)
(2) and (1) refer to region outside and inside the cavity respectively.
Is this just another way of writing:
\left[ \vec{D}_2 - \vec{D}_1 \right] _{r=R} = \sigma
?