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roam
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Homework Statement
Two long, conducting cylinders of thin metal with radii ##R## and ##5R##, respectively, are arranged concentrically so that their axes coincide. The smaller cylinder is placed inside the larger one and the space between them is filled with a dielectric whose relative permittivity ##\epsilon_r## increases with radius in a linear fashion:
##\epsilon_r (\rho) = \alpha \frac{\rho}{R}##
The potential difference between the cylinders is ##V_0## volts.
Use the differential form of Gauss's law to show that the electric displacement D between the plates is:
##D_\rho (\rho) = \frac{C_0}{\rho_0}##
Where ##C_0## is some arbitrary constant.
Homework Equations
[/B]
##\nabla . D = \rho_{free}##
Here ##\rho## is the charge density (not radius as defined in the question).
In cylindrical coordinates divergence is:
##\nabla . v = \frac{1}{\rho} \frac{\partial}{\partial \rho} (\rho v_\rho) + \frac{1}{\rho} \frac{\partial v_\phi}{\partial \phi} + \frac{\partial v_z}{\partial z}##
Polarization is: ##P=\epsilon_0 \chi_e E##
Also ##\epsilon_r = \frac{\epsilon}{\epsilon_0}##.
The Attempt at a Solution
To avoid confusion, for radius I will use r instead of ##\rho##. In the cylindrical coordinates I took only the first term (since we only have a radial dependence):
##\nabla . D = \frac{1}{r} \frac{\partial}{\partial r} (r D_r) = \rho_{free}##
I can then solve the partial differential equation by integrating between R and 5R. But how can I find the charge density ##\rho## when I'm only given ##\epsilon_r##?
(I've found that ##\chi_e = \epsilon_r - 1 = \frac{\alpha r}{R} - 1## but it doesn't seem to help.)
So how can we relate the relative permittivity to charge density? Or is there another way for finding ##\rho##?
Any help would be greatly appreciated.