A device called a condenser comprises two concentric metal spheres, an inner sphere of radius a and an outer
sphere of inner radius d.
The region a < r < b is filled with material of dielectric constant $K_1$, the region b < r < c is filled with
vacuum. An outer region, c < r < d is filled with material of dielectric constant $K_2$. The inner sphere is
charged to a potential V with respect to the outer sphere, which is grounded (V = 0). We may assume that
a < b < c < d.
a)
i) Sketch the situation outlined above, clearly define each region with relevant information.
[2]
ii) Assuming the inner sphere carries total free charge Q, what charge does the outer sphere carry?
[1]
b) Using Gauss' law, find the electric field, $\vec{E}$, at all points for this situation.
[4]
c) Utilising the properties of linear dielectrics, determine all of the bound surface charges present in this
situation.
[6]
d) By considering the scalar potential of the system, determine the capacitance of this condenser system. [8]
Hint - Carefully consider what happens to $\vec{P}$ across the boundary of a linear dielectric.
