Electrostatics - Where does this boundary condition come from?

[JesseC]

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:

$$\left[ \vec{D}_2 - \vec{D}_1 \right] _{r=R} = \sigma$$

?

 

[NateD]

No, it is not the same. The first equation is a boundary condition for the electric potential, while the second equation is a boundary condition for the electric field.